Will approach this from the standpoint of drums, but the same advice applies to amps and other backline equipment.
The best way to handle this would be for the stage manager to get in touch with the owner of the drum kit and clear adjustments ahead of time. If you know who these people are ahead of time, some emails and phone calls can go a long way towards clarifying these issues. Perhaps the toms are particularly touchy, or the cymbals. (For example.)
Perhaps you'll find out that the drum kit is quite similar to what you're used to, or tuned much the same way.
Next, talk to the stage's audio engineer and ask the same question. In addition, mention that you're planning to bring some of your own pieces and warn them that they'll be facing a swap or two. If anything here will cause problems from an audio standpoint, you'll hear about it here.
(Of course, if your music is tom-heavy, or you need a double bass drum and none will be provided, that changes this advice.)
Finding out more about the backline kit
If none of this is possible and you have to use the provided kit, see if you can practice ahead of time on the kit you'll be using. Even a few minutes will be better than nothing! (Failing that, observe the bands before you, or talk to those drummers.)
Which pieces to swap out?
Which pieces should you swap out? Personally, I'd want my drummer to be using their own snare and cymbals (including the hi-hat) if possible. Depending on the size of the room, the snare is likely to be the percussion that changes the audio picture the most.
If all else fails
If you can't even do that, you'll just have to tough it out. Look at your set list and start with songs that have more conventional drum parts, moving to the more complex stuff as you go on.
(Also: Out of all the amps, most guitar platers are more sensitive about their sound than bass players. If you can only swap out one amp, make it the guitar player's amp.)