Bands routinely turn up to festivals using their own FOH and/or monitor rigs, backline, lighting packages etc and throw it on in 30 minute turnarounds. It's pretty standard. We had a 30 minute changeover at Glasto on Friday and in that time they cleared off the previous band, and we put in a backdrop, 3.5m LED tall set-piece flown from the roof, our own risers, half a trucks worth of lights on the floor and all of our own audio kit (consoles, cabling, mics etc). We got it all off again in about 4 minutes afterwards. Festival crews are well used to these sorts of shenanigans.
The key is the preparation and communication - which starts way before the gig when you advance your tech spec and start a conversation with the audio provider about what you're bringing and what you need from them.
If you do that - by the time you load in to the show you'll have already given the techs there enough advance warning that between you you'll have a rough idea of how you're going to tackle it. You'll probably have dropped your rack in place earlier in the show, got it powered up, already scanned for clear frequencies for your IEMs, have a loom/individual cables ready to connect your system to theirs - the more you can do in advance of the 20/30 minute changeover, the smoother it'll be.
If you don't do this and just rock up 10 minutes before your changeover and say "can you plug this in?", you'll likely not be met with enthusiasm, but if you've properly advanced it you'll generally find techs will be plenty accomodating and happy to find a solution that works for all involved.