Just something I discovered when moving station and shipyards. It is fine, and ships will usually move with the stations. However, there were two incidents where it did not move as expected:
The shipyard's operator was in both cases someone who had temporarily joined the ship group to start production, then jumped back out. I received an automated DM with a notification of this in both cases when I moved the shipyard, and it seems the ship remained in its old location while the station itself moved. So you should doublecheck the yard operator before you move, possibly reassigning it to someone you are certain is in the group and has the necessary privs (like the leader.)
In any case, the ships appear to continue production despite being in a remote location.
Alex, I think you're correct about the ships continuing production, however, when the final checks are made I guess one of them is if the ship is still at the location of the shipyard. In that case there could be some problems.