I've done this a couple of times. It's not necessarily easy, but it can be done.

You have three options I believe:

1. use packages from the SnapshotRepository. You'll have to figure out which snapshot you want/need to use, and of course dependencies will not have been snapshotted (ie: dependencies of the ROS packages which are part of the snapshot), so YMMV.
2. build MoveIt from source. See the official documentation, but instead of checking out master (or any of the distribution specific branches), checkout a specific commit (ie: something around the time you mention). It will not necessarily be smooth, as MoveIt isn't the only thing that's changed since that time of course. It might be sufficient to checkout an older version of the moveit.rosinstall the guide makes you download, but you'll have to make sure to get a consistent set of source checkouts across all MoveIt and dependency packages listed in that file.
3. use a MoveIt docker image. See again the official documentation, but instead of using the 'head' moveit/moveit:noetic* image, docker pull a specific older version of it. See #q411417 for a recent discussion about this. Difficulty might be getting hold of the hash(es) of an older image, and of course be aware of all the caveats mentioned in #q411417 (they apply as much to Noetic as they do to Rolling, although there are snapshots of Noetic on the snapshot server).

The snapshot repository would probably be the easiest to use. Note that this will get you an older ROS install, not just MoveIt.

Reverting just a couple packages to their older versions is most likely not possible / a good idea. There is no ABI compatibility guarantee in ROS 1 (nor in ROS 2), and everything gets rebuilt for every "sync" (ie: packages moving from testing to the main repository).

As always when mixing different versions of packages: no guarantees it'll work.