After opening the pause menu I clicked on the achievements tab. As I clicked I noticed a pig moved slightly on my screen. I have since tested this with cows, chickens, and pigs more, so I assume all mobs are affected, though I do not know

The mobs don't seem to be just randomly moving when switching to the achievements tab, they're simply unpausing, for example a mob already moving will continue in the direction they we're going until the reach their stopping point, mind you this takes much clicking back and forth from achievements to done, to achievements to done, etc. as each time only let's the mob move a fractional, but viewable, amount

the best way to reproduce this is to be leading a mob with their food (chickens with seed, cows with wheat, etc.)

1. lead a couple mobs with food, make sure they're still heading your direction when you pause. also be sure to have them viewable with the pause screen up
2. click achievements.
3. it may take a couple of repetitions of achievements -> done -> achievements to get it to happen, as it doesn't happen each time.

after some testing, this actually works with the player as well, if you're falling when you pause and you click the achievements you'll fall more while paused, however it only seems to work once, after which you jolt down then back to place again as if corrected. this can be recreated by

1. jumping
2. pausing before landing
3. clicking achievements

I don't believe this factors in, but I noticed this bug while playing the map captive minecraft, which uses the world border which is set up to expand when getting achievements, I was using vanilla however, no mods