Currently your ships get stored at the port they are landed at, or if not stored at a port they are just gone and you need to reclaim them (more accurately they are still where you left them for as long as that server instance survives, but the information isn't persistent and isn't shared across instances).
There are two exceptions to this - if the game crashes and you re-join the same instance (usually would pop up as a crash recovery option unless the server died), then your ship will be where you left it when you disappeared. Secondly, if your ship has a bed and you logout using the bed then when you log back in you will respawn inside your ship at the location you were at when you logged out... except when it doesn't work. Was pretty reliable for me in 3.7 though and haven't really played enough 3.8 to comment.
If you landed Ship A at olisar and then took ship B to Hurston, you will find when coming to request it at the ASOP that Ship A is stored in Olisar. In order to request it in Hurston you would need to file a claim and wait for the countdown/expedite (or fly back to olisar and request it there of course). The big issue with doing that is if you have eg cargo or valuables in the ship as you are effectively getting an insurance replacement minus the contents.
Longer term will be more similar to the same instance/bed logout scenario - the servers will persist the location data across them and across sessions so where you leave your ship is where your ship will be until it is moved, claimed or destroyed. There is then of course also the question of hangars and how they will come into play, but one assumes they will effectively be like private versions of port hangars at your own locations.
