Today I explored the new features of adjusting movesets from within HOME to see if it would really affect anything in PLA. While PLA let you rearrange moves freely, you can't delete them once you have 4, and also there wasn't a way to have moves in your moveset that you don't have learned before (aside from that thing that got patched where the alphas who spawned with a tutor move didn't have it marked as learned). Overall I wasn't expecting much but it does add some functionality that you couldn't do in PLA before.
I took a Level 35 Flareon from PLA and into SV to obtain the (future at 43) level up move Double-Edge and tutor move Fire Fang, both available from the SV level up learnset. For good measure I also deleted the other 2 moves in HOME to see if anything would happen.
I was wondering what would happen if you had moves that you didn't know before in your moveset and swapped them out. The answer is that the moves don't show up in your move list and just disappear when you swap them out.
I also tried to see what would happen if you mastered a move, forgot it, and then learned it afterwards. It turns out that it stays mastered when you relearn it again. At the same time I noticed that you can also have the choice to be taught a move that is currently in your movepool, though this doesn't really change anything because PLA doesn't automatically fill in your movepool either.
While going through this, I noticed in HOME that Flareon had Mimic listed as a mastered move, even though it can't be mastered. This is how it looks in PLA itself too, and I guess this is because in the move data it's listed as being mastered at Level 0. Then I looked through for the other unmasterable moves (Teleport and Splash) and noticed that Ralts was listed as mastering Teleport at 15. Abra didn't have this, so this is just an error for the whole Ralts line.
I decided to see if I had any Ralts to compare with, but they were all above Level 15 already. I checked serebii's map filtering for Ralts, and every single spawn I found was minimum level 16, so now my plan was to bring over a Ralts to compare the two and see if it would say that it mastered Teleport.
I took a Level 12 Ralts from SV and moved it to PLA. As expected, it had the unmastered Teleport icon at this point. I raised it to Level 15 and it mastered Teleport.
I also tried to use a Seed of Mastery to get a screenshot of Ralts mastering Teleport by name, but Zisu only lets you master moves that she has in her tutor pool and not natural level up moves, so it didn't work.
So what did this mean in the end? Well, you couldn't legally see a Ralts with an unmastered Teleport in PLA before HOME because all of them seem to spawn above Level 15, so I think this might be the first time that anyone has seen Ralts master Teleport in PLA for whatever it's worth.
In the middle of this I noticed that Bulbapedia says that these moves can't be mastered, but we've just seen that Ralts can go from unmastered to mastered Teleport. I guess they should say that they can't be used in battle with different styles instead since they show up with the mastered icon in every other case, except Struggle, though then you could argue the fact that you can't see Mimic or Splash be mastered in a dialog box anyway.
In that sense Hidden Power would also be a move that you can't ever see mastered normally in PLA since they all start at a minimum of level 25, but if you brought a level 5 Unown from Johto and leveled it up to 10 in PLA, then you could see Unown master a move in PLA, which is also something probably no one has done before. So there's some more fun new activities you can do with HOME if you have nothing better to do.
tl;dr PLA doesn't add HOME moves to your movepool automatically but it remembers their mastered status even if you forget them, and I also found out that Ralts masters Teleport at a non-zero level lower than it naturally spawns at and used HOME to see it happen for the first time.