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As long as it's warm enough for them inside the coop, I wouldn't worry about a draft. It's the cold drafts in cooler seasons that can be problematic for chicks.
If you run a heat lamp to your coop PLEASE, PLEASE, PLEASE be sure to secure it in at LEAST two, if not three, different ways, so in case one fails the others will prevent it from causing a fire.
I have very successfull used cardboard boxes for many, many batches of chicks. I start with a large box, line it with doggie piddle pads (b/c of the absorbent padding, if there's a water spill it will save the cardboard from being ruined). In the first week of the chicks' lives, I line the top of the doggie pads with paper towels (to avoid spraddle leg. Never use newspaper.) Then, I switch over to pine shavings on top of the piddle pads after the first week and change them as needed.
When the chicks get bigger and need more space, I make them a "chicken condo" by putting two cardboard boxes together with duck tape and cutting out a "window" in between the two boxes (lining the window with duck tape so shavings don't get in between the two boxes) for them to be able to walk through. I usually leave the heat lamp on one end and the food/water on the other end of the condo. When they need more heat, they'll congregate near the lamp, when they don't, they won't. I like to put a little roost into the box as well so they can get practice for their big chick days. It should not be round though, they're not parakeets.
Add more large cardboard condo rooms as needed.
You will know when it's time to put something on top of the brooder to prevent them from flying or jumping out. I use window screening. You may never reach this age when you need to cover the brooder b/c they may be ready to live in your coop. I have used this system primarily in the dead of winter when it's too cold out even for 6-8 week old babies.
It sounds like you should be good to go with your coop though, so if you can get them out there, that'd be ideal. They don't have to be confined to a box inside the coop (as long as they're the only chickens in there). Just put the heat lamp in one corner and they'll go there if they are cold.
It's a good idea to keep chickens of any age confined to the inside of the coop for a week or two before letting them out into the run. This accomplishes two things: #1) it reinforces the expected location for roosting at night (so you don't find yourself having to chase them to go in at dusk) and #2) it familiarizes them with the location of the nesting boxes for future use. After a couple of weeks, they can go into the run for another week or two but don't let them free range yet. Again, it reinforces going into the coop at night. I always use this technique for new additions to the coop and it works amazingly well.
Good luck and have fun with your new babies!