In "Flesh and Blood," the holograms beam the EMH over to their ship. Just prior to doing so, Tom Paris attempts to download the EMH's program into the mobile emitter. I assume this would keep the EMH "grounded" on Voyager, since he could simply be "booted" up in sickbay using the mobile emitter. I think that for the hologram ship to claim the EMH, the holograms would then have to beam over the mobile emitter, and presumably this is not easy.
However, when the EMH is beamed over, how can he remain intact? Essentially the EMH is nothing but light and forcefields, and if his program is still in Voyager's main computer, how could he be "grounded" on the hologram ship?
Unless, perhaps, the EMH program in Voyager's computers was powering the EMH from a distance. Though when the hologram ship went out of range, the EMH wouldn't be able to remain "intact" any longer.