There are two ASDS ships.
Just Read the Instructions (JRTI) which was first deployed on the Marmac 300 hull, never successfully landed a stage, had its wings removed and moved over to a newer hull (Marmac 303 I think), then shipped through the Panama Canal to the Pacific.
In the Pacific, JRTI has landed the Jason-3 and Iridium-Next first cores as of Jun 2017. (More Iridium missions in 2017 to come).
In the Atlantic, Of Course I Still Love You (OCISLY), based on the Marmac 304 hull was deployed and has landed multiple stages.
The July 17th, 2017 planned launch of BulgariSat on a previously owned (gently used, mission tested, pick your euphemism of choice) Falcon 9 first stage, (was used in the Iridium NEXT 1 launch that landed on JRTI in the Pacific) will try to land on OCISLY in the Atlantic.
This will make it the first stage to land in both oceans if (when?) successful.
Moving them ocean to ocean would be a lot of work. The Panama Canal is only wide enough for them if the wings are removed. They tow slowly, so it will take a long time.
In all likelihood if they get the launch rate up much higher they will soon need a second ASDS in the Atlantic. If the Mod 5 updates, with the 'fullest thrust' engines allow enough margin for RTLS even on big GEO missions, that prediction could be voided.
They have developed an unused (as of this time) Roomba like device to try and automate grabbing and holding a landed stage. Cannot wait to see that in action. But the goal there is to enable quicker turnarounds, and even enable turnarounds in bad ocean weather where it would be unsafe to load a crew to secure the stage.