Whenever an enemy ship downloads something from the board computer or whenever a hologram gets transferred, the data gets removed from the board computer. It's almost as if the data was a physical object. Why doesn't the original data get preserved like with current computers? It seems so illogical to do it that way.
Voyager 6x24 - The doctor gets transferred to the alpha quadrant in order to treat Louis Zimmerman. The Voyager doesn't have the EMH for that time period
Voyager 7x05 - The Doctor's mobile emitter gets stolen and replaced with a replicated copy. The original Doctor isn't in the board computer anymore
Voyager 7x06 - Barclays hologram gets intercepted on the way to the Delta quadrant and you can hear on multiple occasions in the episode that the hologram is gone.