I assume you're asking about treating for varroa mites. because swarms don't bring any brood with them, mite loads are typically small to non-existent in a new colony, so there's no reason to treat for a while. allowing your colony to swarm also takes advantage of this because there's a period without any brood in the hive while the new queen is getting up to speed.
there are a few ways to monitor mite numbers. I generally spend a few minutes from time to time watching the entrance. if there are a lot of mites, you'll see them on the landing board and on bees coming and going.
I'm with Mike, though: I don't treat.
I don't know anything about swarm season in Wyoming, but it's getting to be pretty late where I'm at. a late July swarm would have a pretty difficult time building enough population and putting up enough honey to survive the winter here. that isn't to say they wouldn't make it, just that their odds would be a little longer. and if they didn't, they would still build comb that the next swarm to move in could use to its advantage.
If there are many beekeepers around you then no-treating is not really a nice thing to do because while performing natural selection on your bees you will also be a varroa breeding ground for your neighbors. Varroa being inserted in your hives by somebody else's bees on a robbing mission is very real.
At our place we make do by treating with organic acids - formic and oxalic. I prefer that since those are chemicals that are naturally present in honey although, of course, not at the concentration that we use for varroa control.
Since the hive has been out of use for a while formic acid would give you a double benefit: cleaning the newly resident bees and the hive itself since formic has been known to take care of various molds and fungal based disease. Not just in theory, that's been my experience as well.
Read up on what's recommended for your local enviroment - for us in Slovenia, a long way away from you though also in a temperate climate, a single treatment is 20 ml of 65% formic per box, dripped onto a sponge or similar material when daily temps are not above 80 F or so - and repeated 2-3 x with a week's pause inbetween.
As to what was left in the old hive before the new swarm came -- normally it's a good idea to clean hives with a burner (or a bath in a caustic solution) because varroa, the arch enemy, is not just a bug that does vampiric stuff to your bees - it also leaves a hole in their body which won't heal. This can lead to the situation where a heavy varroa infestation also creates a breeding ground for various viral, bacterial and fungal disease. Thus the fire / chemical thing.
Elle, it's not my aim here to discourage you, I'm just somebody who doesn't particularly like risk and thus the precautions above. Your hive can be entirely OK. It already has new tenants anyway so here's hoping they will thrive.
One more thing, you mentioned that the hive is full of comb. Now, a swarm is a comb-building machine and it's a good thing to let it do its stuff. It will keep the bees in good form and you'll get fresh wax. So in my opinion it would be a good idea to remove not just some of the frames which contain honey (for later use) but also some of the non-honey-filled old comb - preferrably the darkest, ie. most heavily used one by your previous bee family. That way when your new queen starts laying it will happen in a new, healthier environment.
Be safe when using organic acids! They are not harmless in their concentrated form,. Read about usage. Protect your lungs, eyes and skin. If you dilute your acid, pour acid into water, NOT water into acid.