Okay, this is a lot. In order to actually be able to answer it without writing an essay, I won't make complete comparisons for each subrace. As I mentioned in a comment, if you want a detailed feedback, I would create a separate question for each subrace - they are too different from each other.
Your Base Race has almost no features, which is fine considering the main thing is on the subrace. It's still worth to note that consistently the base race is the one giving a +2 AS bonus.
Earth
You get a +3 AS, which is the usual for playable races. +2 Str +1 Cha is an awkward distribution though (same as dragonborn), probably most useful for Paladins (usually the class Dragonborns are played). For most of other classes, one of the two stats will be kinda wasted. By itself, this doesn't make the race weaker. Tiefling gets an even worse bonus distribution.
Barkskin is a 2nd level spell. Not only you are giving a free 2nd level spell to the character, you are giving a free 2nd level spell slot as well. We could enter the discussion if barkskin is a strong spell or not (my opinion: it's not, actually it's weak), but the fact is: 2nd level spells to 1st level players is alot.
I would change it to a feature that might scale with PC's level, similar to Breath Weapon from Dragonborn.
How strong will your resistance to piercing be depends on the campaign. While Dragonborn and Tieflings have stronger resistances (elemental), they are more useful in later stages of the game, while piercing might be too strong in early stages, where physical damage is most common.
I would say the Earth subrace is (maybe too) strong in Tier 1 adventures, but becomes balanced from Tier 2 onwards. I'm not sure about Tier 3, but it becomes kinda weak in Tier 4, compared to the playable races.. One easy way to fix it, as I mentioned, is making the Barkskin feature scale with the tiers. I will make a note here that is valid to the following feedbacks, though: The bonus from races usually are less important in the endgame anyway, since your class features and magic items will compensate for it.
Air
Similar to Aarakocra. I am not sure the +10 move speed in land and Hover compensate both the -1 Dex and -20 Move speed flying, though. Also, I'm not sure how mechanically useful the weather prediction thing will be, that depends on you. Also, while Aarakocra +2 Dex +1 Wis helps alot with Rangers and Druids, the +1 Cha instead makes it a little more awkward, probably being more useful for Bards. Sorcerers might be interested in the flying feature, though, and get more from +1 Cha than +1 Wis.
As it is, it's clearly weaker than Aarakocra. (Yes, I know you asked "compared to PHB", but since PHB doesn't have flying races, I'm forced to compare it to EE, sorry). That said, it isn't completely underpowered since the AS bonus is more attractive for bards and sorcerers.
Fire
My first impression was god it is too strong. Then I noticed that, while you added a lot of powerful features together, they don't have too much synergy.
The main problem is that, from internal comparison, unless Flying can compensate it very well, it's way stronger than the Air version.
Other than that, it's very similar to a Red, Brass or Gold Dragonborn, changing the Str for Dex.
In my opinion, having the Fire Bolt is weaker than the Breath Weapon, even if it can be cast as many times as you want, because you probably will have other ways to deal damage anyway. Thinking in Dex based characters, they will probably be shooting arrows instead of fire bolts. The Breath Weapon is AoE, though, and usually cast by a class that doesn't have that much AoE early (Paladin), so it's more useful in the specific situation it's actually used.
The Grappling damage would also be more useful if it was a +2 Str bonus instead. Building a grappling character without a Str bonus is too suboptimal. I can't even guess how strong this feature will be as it is right now, sorry but you'll have to playtest.
So, comparing to Dragonborns, it will depend on how much Darkvision is a think in your setting. If darkvision is really important, it's balanced compared to Dragonborn.
If we compare it to Tieflings, it seems to get outscaled. Early on, you have a damage cantrip and a grappling feature, while Tiefling has a flavor cantrip. From level 3 onwards they get spells, though, while the scaling of this subrace is only based on the fire bolt cantrip. Again, even if it gets outscaled, it's not an obvious choice since your AS bonus is more attractive to Dex users.
Still, I would fix the fact that the AS bonus (dex) and the features you are giving don't synergize well.
One possible solution is to change the Fire subrace to +2 Str and the Earth to +2 Con instead (if you don't want both to have the same AS bonus). It would fit the "durable" flavor of the Earth/stone and the "aggressiveness" of the Fire.
Note that, if you do, the Fire subrace becomes a very strong option for grappling buildings. I wouldn't say this makes it overpowered, though, since grappling builds are usually more for fun than optimal.
Water
This one is hard to compare. The Healing feature seems weak (just Your Level is not much HP. Compare with Dwarves getting +1 HP/level, which is arguably stronger). Create/Destroy water is a 1st level spell, but this one doesn't need much discussion about being strong or not - it's not strong mechanically.
So it's about the other 3 features. First, the AS bonus: +1 Con +1 Cha is weak. Variant humans get to choose their two +1s and get a Feat on top of that. But they are arguably the best race in the game, so let's not do this. Still, +2 AS is underwhelming and I would expect the features to compensate it strongly. For the Air case, flying might do it, let's see the Water.
Water Breather again will obviously depend on your setting. It's arguably harder to use than Darkvision or Flying, though, since while your party can easily fight in a dark cave or an open field, fighting under water is way more unlikely and usually not desired by PCs.
That leaves Armor of Agathys to compensate the underwhelming AS bonus. Well, sadly it doesn't. It is a very, very strong feature in the first two levels, where +5 HP and 5-10 damage to the enemy are alot, but it quickly becomes weak as you progress in the game. Again, I feel you are lacking scaling. Increase these numbers as the tiers go up. So, similar to the Earth subrace, I'd say it is (maybe too) strong in Tier 1, but gets weaker (and even underpowered) in Tier 3+.
TL;DR
My main criticism here is that all your subraces lack scaling. While Air is fine without it, since Flying itself has a constant utility through the game, the other subraces might suffer from it. It is fine if the campaign is focused in Tier 2, but they seem too strong for Tier 1 and too weak for Tier 3+. Again, note that being too weak in Tier 3+ is not that important, since you have lots of class features and magic items to compensate for that. Still it's something you might want to take into account.
I suggest scaling as a way to fix it, but introducing mechanics that have utility through the game (such as having advantage on something, proficiency on something, etc) also works.
It's worth to mention, though, if you are playing a long campaign that goes through, for example, levels 1 to 15, your subraces are mostly fine, as they will be strong early on and weak later, and it's a valid trade-off for your players to choose. I have linked a chat discussion about this, if you are interested.
Side note:
If you are wondering why I'm not comparing to Dwarves, Elves, Gnomes or Halflings, which also don't have numbers scaling with the levels, it's because they have proficiencies in skills, weapons or tools, or advantages in saving throws, which are the things scaling. Also, they get lots of other features, such as Trance, Luck, etc. Comparing to Dragonborns, Tieflings and Aarakocras is easier and more fair (imo) because they are closer to what OP has made. It's hard to compare if proficiency in perception, using daggers and sleeping 2 hours less is better than 1d8 damage when grappling.
Best Answer
It looks balanced but you should test it on a monk
I normally would use the Detect Balance sheet to score this. John Doe already did that, and it looks balanced on the surface.
However, there is one issue with point-buy scoring systems like Detect Balance: they tend to break down if there is strong synergy between multiple features. The flat scores cannot capture these interactions. Of course, finding synergies is half of the fun of building characters (or Magic: the Gathering decks, if you are into that - where combo decks demonstrate how individually innocous elements can combine to create something that can be game-breaking).
All the "special" features in this build already work very well together. You can use Dex for unarmed melee, shoves and grapples, you get extra mobility with Nimble Dodge on such melee, and you get free Shove actions on hits. They make for a nice pseudo-monk. The free shove action, which normally would cost you a full separate action looks like the strongest element, and having it gated by proficiency bonus uses helps.
The real question is how this will work on a monk. For example, the Nimble Escape helps to conserve Ki when you do not need to Step of the Wind to Disengage from an enemy to move away. As you pointed out, you can use Trip and Bash in combo with the monk's unarmed attacks. Shoving them also gets you out of close range or can be used to drop your opponent prone, a strong effect if used with Flurry of Blows where all the additional attacks then happen with advantage while he is down.
I think this is still fine, but it's these kinds of sublte advantages piling upon each other that can become dangerous, and I would recommend to test this, and be ready to nerf it, in case it turns out too strong. (In particular, if you want to make this available to players, discuss this is experimental and have the player understand there may be some correction later on if they pick it and it turns out unbalancing.)