Saddled horses with NoAI are affected by motion when riding them after they have been attacked or pushed

This bug remains in snapshot 18w07a.
And also, when you ride a horse, a donkey or a mule with saddle and NoAI, it is affected by gravity and falls to the ground.

Confirmed for 1.13.1 release.

Like @unknown already commented, if you summon a Horse with NoAI e.g. in the air, it will not drop down (= intended), but as soon as you mount it while in the air, it will drop down.
It doesn't matter whether or not it got a saddle, it only has to be tame.

Also, if you set any pitch value and it is saddled, it will lose that pitch value when you mount it, no matter if it was in the air or not.
It will keep its pitch value if it was not saddled though.
Can be tested with:

/summon horse ~ ~ ~ {NoAI:1b,Tame:1b,Variant:0,Rotation:[90F,-90F],SaddleItem:{id:"minecraft:saddle",Count:1b}}

Furthermore, horses with NoAI set to true will still wiggle their tail (WaI?).

Lastly, as long as it is saddled and tame, you can ride a NoAI-horse normally and also make it jump. Unsure whether this could be a keeper though, as it may enable mapmakers to some neat contraptions with a controllable mount that is otherwise non-moving. So before this would be changed, it'd be great if Mojang could discuss this internally, preferrably also with mapmakers.

Confirmed for 18w49a.

Relates to MC-107849.

Confirmed for 19w13b.

Confirmed for 1.14.2 pre-3.

Affects 19w37a

Confirmed 1.15 pre2. You don't need to punch it, it drifts in the direction you look when you ride it.

Can confirm in 20w48a.

Can confirm in 21w03a.

Can confirm in 21w05b.

Can confirm in 21w06a.

Can confirm in 21w16a.

Can confirm in 1.17

Can confirm in 1.17.1.

Can confirm in 1.18.1.

Can confirm in 1.18.2.

The behavior of this problem has changed, so I've updated this ticket accordingly to reflect the new behavior.