Where I have seen this behavior, it has been due to the masonry walls "delaminating" (i.e the exterior with is coming apart from the interior wythes).
Usually it is a 3 with thick wall on an early 1900's or late 1800's building.
Usually it is due to water intrusion and many, many freeze thaw cycles.

Curious what city you're in.

Very dangerous to take the wall down but then... kinda dangerous to leave it up too.

It is in the historic part of Baltimore. Now could that actually be bulging all out due to a possible absence of lateral restraining on the 2nd floor, in which case it may not be just one wythe moving outwards?

That stone doesn't look historic but the pictures aren't from the right angle to see properly. Are you sure it isn't 1970s "updating" of an old building by sticking masonry veneer and/or stucco over an old masonry building and now it's falling off? That's a pretty common problem that I've seen in a lot of places.

It looks like a product I have heard of as "Permastone" .
It was sold as an upgrade to brick exteriors, maybe 1940's +/- (not sure but the seems about right based on the buildings I've seen. Might be earlier though. I have seen it on churches (as a way to make them look more "classical" ) and many row house type structures.

I have been told they are VERY common in Baltimore but I have seen it in many cities Midwest cities. My son moved to Baltimore recently and I have seen several houses there with this condition.

I agree that something is "delaminating" but I would be curious if it is just the "stuck-on" portion or if the outer wythe is coming off with it.

If you get to the bottom of it, I'd be interested to hear what you find.

It is an historic area. That doesn't mean that in the 70s and 80s they had rules, or in a city as messy as Baltimore, abiding to rules in those times might have been just utopian anyway. But you guessed it right! Definitely, these are cheap veneers that were added a few decades ago to cover problems, and these buildings are not quite exemplary of what historic Baltimore can have; Not that it compares to historic Paris or other historic cities have to offer.

I also wonder if it is bowing of full masonry walls (inner and outer wythe together.) If the wall is not connected to the upper (2nd & 3rd) floors, it may be standing upright and restrained only in one direction. It cannot bow toward the interior but can bow outwards due to the fact that there is no restraint. It would probably deform that way and the layout of the bricks would adjust to the vertical stress. I am just sharing a thought here.

My approach would be to remove some of the stones that are bulging out to determine the cause however, you may require a sill plate to support the stones over the opening before totally removing the stones.

I wouldn't want to remove anything until I was confident about the stability of the existing/remaining portions. Usually the interior walls are furred out so it's hard to tell without some demo inside.
Looks like the floor framing would be parallel to the front walls so I'm not too sure how well it is anchored.
From what I've seen, early 1900's brick construction using metal anchors for the outer wythe was just getting started and I don't think corrosion protection was very common. Plenty ov masonry was built with header courses still but some was not and I suspect much from this era in Baltimore is in the latter category.
I don't know how well the "solution" of using Permastone really is. Obviously it has lasted a good long while but now may be the "end of the road" for it.
Overall just a risky situation (IMHO).