[RPG] Does Mask of Many Faces / Disguise Self move with the target

Strictly better? No.

One of the major points of Disguise Self is that it can not only alter your appearance (via illusion), but your clothing and equipment as well.

It's important to note the inverse of this: Alter Self does not specify that it modifies your clothing or equipment. This means that, depending on how severe your alterations are, you may no longer fit into your armor and your clothing may clearly no longer fit you, depending on how simulationist your DM is on the topic. This is especially important if using the "Variant: Equipment Sizes" rule on PHB p.144:

In most campaigns, you can use or wear any equipment that you find on your adventures, within the bounds of common sense. For example, a burly half-orc won’t fit in a halfling’s leather armor, and a gnome would be swallowed up in a cloud giant’s elegant robe.

The DM can impose more realism. For example, a suit of plate armor made for one human might not fit another one without significant alterations, and a guard’s uniform might be visibly ill-fitting when an adventurer tries to wear it as a disguise.

Using this variant, when adventurers find armor, clothing, and similar items that are made to be worn, they might need to visit an armorsmith, tailor, leatherworker, or similar expert to make the item wearable. The cost for such work varies from 10 to 40 percent of the market price of the item. The DM can either roll 1d4 × 10 or determine the increase in cost based on the extent of the alterations required.

Even if your equipment does fit you after the effects of the spell, any well known gear or accessories may still let you be identified, especially if they're more well known then your actual physical attributes.

There are a few other considerations as well.

One of them is that both Alter Self and Disguise Self can be used at the same time, to enhance the illusion with actual physical changes underneath it. Do note that this is only possible because Disguise Self does not require concentration, which is another point in its favor over Alter Self, which does. This means Alter Self can fail earlier than intended (especially in combat or while under stress), while Disguise Self will last the duration under most conditions, and still allows you to cast other concentration spells.

Another is comparing how Disguise Self and Alter Self interact with height & size.

Alter Self actually changes your height, while Disguise Self only creates the illusion of a height change, keeping you your usual size. This might matter if your DM actually cares about character height in certain situations rather than just your size.

In addition, Alter Self restricts you to only changing your appearance to a creature of the same size with no other restriction on maximum height gain/loss, while Disguise Self only has a restriction on height change rather than size change.

For example, with Alter Self, you could change the shortest possible dwarf into the tallest possible goliath, whereas with Disguise Self, you could make a dwarf appear to be a halfling (so long as you're not trying to change your height by more than a foot in doing so).

Overall, I'd probably rate Alter Self better than Disguise Self in some combination of the following situations:

• There is plenty of time to make preparations in advance, including having an appropriate alternate outfit ready.
• You are unprepared, but you only need minor superficial changes with no drastic change such as size, making the lack of alternate costume not an issue.
• You are in no danger of having your concentration broken for the next hour, nor are you likely to need to cast any other Concentration spells.
• You are likely to be thoroughly inspected for some reason or another.

In other situations, there is a chance Disguise Self may be the more appropriate spell. In a situation requiring an emergency drastic appearance change (clothes, size, and all), for example, I would much prefer Disguise Self as my option.

A Big Kobold's OK

The Player's Handbook p 222 (disguise self) says:

You cannot change your body type. For example, a human caster could look human, humanoid, or like any other human-shaped bipedal creature.

The example, which is not in itself rules, gives a reasonable answer that body type is 'general shape and number of limbs'. It also calls out humanoid (apparently the type, considering the later clause regarding shape) for humans. A kobold is humanoid, so a changeling could reasonably become a larger than normal kobold (given the height change limitations of plus or minus one foot in the spell).

How It works For Changelings

For the complete limitations of the ability (see the Minor Change Shape sidebar in the link), as it relates to changelings:

• You can [become] 1 foot shorter or taller,

(Simple enough)

• thin, fat, or in between

(Conservation of mass not being a thing in D&D... No real limitation listed; though flesh appears to be required, so no skeletal creatures)

• You cannot change your body type. For example, a [changeling] could look human, humanoid, or like any other human-shaped bipedal creature.

(This is extraordinarily broad; definitely any humanoid, most monstrous humanoids (young minotaurs, medusae, large dromites), plus many fey, several devils and demons, many celestials and a significant variety of other outsiders, some giants (like half giants), some aberrations (like elans), just about any undead, and perhaps even a medium ape or baboon)

• Otherwise, the extent of the apparent change is up to you. You could add or obscure a minor feature or look like an entirely different person. The [ability] does not provide the abilities or mannerisms of the chosen form, [but it alters your actual] tactile (touch) and audible (sound) properties [though not that of your] equipment.

(You have extremely broad leeway, and you look, and feel and sound like the form you've assumed. The sound properties are demonstrated as Rules As Intended at the very least in the Eberron Campaign Setting and Races of Eberron sections regarding changelings, such as RoE p42-4: 'A Day in the Life' section.)

How It works Otherwise

As for the limitations of the disguise self spell itself, the only differences from the above are that the caster only appears to have the listed changes, and its equipment is visually disguised (so no effect on move silently, for example).