With the EZ 3in1 it works in conjuntion with a Slot-1 flashcart. The GBA ROMS are stored on the memory of the Slot-1 card (built-in or on Micro SD) so you can store as many as you want on your SLOT-1 device as long as you have the room for them.
Now using the NOR memory of the EZ 3in1 card you can only flash 1 game at a time to it up to a 32mb ROM. It will stay there until you flash another game to NOR in the 3in1. This means next time you boot it will still be in NOR memory, you can even carry the 3in1 to another DS and the game can be played off SLOT-2 without a SLOT-1 card.
As for PSRAM, again you can only flash 1 game to it at a time, but as soon as you power down the DS you loose the game since it was stored in volatile memory. You do not loose it on the SLOT-1 card but next time you boot it has to be loaded again to the 3in1's memory in PSRAM mode.