There are numerous places in various Star Trek episodes where references are made to the fact that "downloads" are a destructive operation: you download something from one place, and unless you explicitly made a copy beforehand, what you downloaded is gone from its original storage location and instead appears in some other location. Hence was born the term "copy and download".
There are of course out-of-universe, plot reasons for things happening that way which are exploited from time to time. There is also the out-of-universe explanation of how large media companies feel about people downloading copyrighted works, and a possible out-of-universe explanation that at the time, people in general may not have been intimitely familiar with the implications of the term "download", so they added "copy" to help describe what happens to at least portions of their viewership.
But is there a plausible in-universe explanation for why downloads would be destructive by default, and a specific copying operation would have to be made beforehand so as to not delete the data from its original location?
As for specific examples, these are two that I can think of right now (I'm quite sure there are more):
- VOY "Twisted" (S2E06), right near the end, stated by Torres: "Our entire database has been copied and downloaded into somebody else's system."
- VOY "Message In A Bottle" (S4E14), a little more than five minutes into the episode, the EMH is "downloaded into the transceiver array" for transmission across an alien sensor network to the Prometheus when the latter is in the alpha quadrant, and some 20 minutes into the episode it's established that the EMH is not available in the sickbay and will return later.