In my most recent ToA game I was building and giving ships to the Altarians, who needed these ships very badly as they were losing a war.

I'd launch the ship (right next to my planet) and then I'd give it to the Altarians; it's color would change to Altarian blue. But then several turns later I'd notice that it was still right next to my planet. Why won't they move their new ships?

Eventually I stopped giving away ships, but only for this reason. Can anyone please tell me if there's anything I can do to make races whom I've given ships to move their new ships? Thanks in advance

I never noticed that; could it be that they were suddenly out of their range, i.e. too far away from any Altarian planet or starbase?

I have this problem too... Sometimes they move, but usually they just sit there. You can move your ships to your territory when they are transported there by a wormhole, so this is probably just a bug with the AI.

I've never seen traded ships sit in place like that.

The only ships that should be transported by a wormhole are survey vessels that have enough range to move on the map. It's other ships with limited range that might be out of range when you hand them over to the AI.
The solution would be to send them to the border between your sphere of influence and that of the AI before handing them over.