I bought a house recently that came with a coop. I've been cleaning it out and prepping it for my pullets but haven't been able to get rid of the mouse family living in there.
I don't expect traps to eliminate mice around my coop. They live all around in hay fields and pastures. I use traps to reduce the numbers to manageable levels.
In your situation I would have used poison to try to eliminate any nearby before you were ready to take the chickens out there. Even that won't totally eliminate them but can quickly bring the numbers down. It may be too late to go that route.
The chicks are getting too big for their brooder and I really want to move them out to the coop but worried it would cause health issues with the mouse poop and potential fleas?
There are health risks with the mice and their poop. There are health risks with chicken poop if you let that build up too thick. My goal is to keep the numbers down to minimize those risks. They can also eat a lot of expensive chicken feed.
Or would the mice move out once the chickens start going after them?
Some chickens are better mousers than others. Some can catch them, some don't even try. Many chickens will eat a mouse if they can catch one. If one gets one you can see a game of tag as they all try to take it away. But it is unlikely your chickens will ever eliminate them. The mice can find nesting places where the chickens cannot get to them. They come out at night when the chickens can't see them.
I made an Adirondack bucket mouse trap similar to what Azygous is talking about. It does catch mice. You need enough water in the bottom so the rodent cannot touch the bottom and jump out. No poison is involved so I feed those mice to the chickens. I do not put any traps where the chickens can get to them but have a shed where the coop is on one end. I put the traps next to the coop in that. Raccoons and possum like peanut butter too. I have had my trap messed up when one of them went after the peanut butter.
I also use snap traps in that area. I attach the snap trap to a screw eye with a wire so the trap isn't carried off if a larger animal gets caught.
Chicken feed is a big attraction to mice. I store my bags of chicken feed in a metal trash can so the mice cannot chew their way into it. Putting the feed dishes away at night will not attract mice as bad. Some people only feed in the run to avoid attracting mice into the coop after the feed but they will be attracted to the run if you keep feed out there.
I don't know how you can totally eliminate mice, they have been around humans for thousands of years. I take measures to reduce the risks and damage.
Good luck!