Grappling doesn't care about weight, only size matters.
The grappling rules do not distinguish from grappling a 300lb Medium-sized creature from a 100lb medium creature. If you're grappling a medium-sized princess, it doesn't matter how heavy she is, your speed is always halved, unless the creature is two or more sizes smaller than you.
So, in your example, the Barbarian could only move 25 feet then dash for another 25 feet. And, yes, they can do 5d6 damage, if everything lines up for them. On subsequent rounds, the Barbarian could just move back down, grapple again and dash up 25 feet. Equating to 2d6 damage on subsequent rounds, not that impressive any more.
It should also be noted that carrying/lifting/dragging an object is governed by different rules from carrying (i.e. grappling) a conscious kicking screaming princess.
Yes, it applies to all movements
Unarmored Movement (PHB, 78) States:
Starting at 2nd level, your speed increases by 10 feet while you are not wearing armor or wielding a shield.
It does NOT qualify as your Walking Speed (or Swimming Speed, or Flying Speed)...just Speed. This applies to all types of movements and would add additional speed to the Fly spell as well.
This is supported by Jeremy Crawford as well.
Unarmored Movement is intended to increase a monk's innate speed, including an aarakocra monk's flying speed.
He further clarifies Bonuses and Penalties that apply to Speed in general:
Bonuses/penalties to speed apply to your speeds in general, unless the text specifies walking, flying, etc
As the Unarmored Movement does not specify a specific type of speed, it applies to all.
Fly Spell specifically
Fly states (PHB, 243)
The target gains a flying speed of 60 feet for the duration.
The implication here is that you (the target) had no Fly Speed (or a lower Fly Speed) before casting and it is now a speed that you, the target have. Once it becomes your speed, it becomes subject to Unarmored Movement.
Similar to Special Types of Movement in the PHB, page 182)
While climbing or swimming, each foot of movement costs 1 extra foot (2 extra feet in difficult terrain), unless a creature has a climbing or swimming speed.
Here, Swimming has a penalty, unless they have a Swimming Speed. Once they have that Speed, from whatever source, then Unarmored Movement should apply.
Rule of Cool
And it's fun! Monks are supposed to be super quick. Let them be super quick in all their speed!
Best Answer
Presumably, their backpack is designed so that it allows for flying.
The only restriction the Aarakocra fly speed poses is medium or heavy armor.
So as far as mechanics of the game go it is possible.
They are an intelligent race with a society so they probably had to deal with this problem at one point. It makes sense that they designed backpacks and other equipment to be worn while flying and depending on how common Aarakocra are in the setting, stores could easily carry equipment tailored to specific needs of the race similar to any other races.