Red mites don't live on chickens so its highly unlikely your new hens bought them in. They come off of trees and move into hen houses. So spread from wild birds usually or if your housing is kept under or near trees and hedges.
Are you sure its red mite? Have you actually checked your housing at night and seen them? You need to go out when its dark, take a torch and check under the perches. Run your finger along under the perch and if fresh blood is there then yes, I'd say red mite. You will also be able to see them running around on the perches and on your birds legs then. And if the infestation is really bad then you'll see them on the sides of the house and so on and be able to see where they are collecting. You won't see them living on the birds during the day.
As Davidd said, serious blood loss will make your birds anemic so their combs and wattles will look very pale. If they are still red then I'd say they are not losing so much blood (or any at all if no red mite).
If it really is red mite I'd go straight for the big guns and go get some proper agricultural creosote (not the stuff they sell in garden centres but the proper hardcore stuff from agricultural merchants) and give the entire hen house and run a good coating. Ensure you move the birds out for at least a week (or longer if possible) to give it time to air out fully. I know some people say move them back in after 24 hours but this is nasty stuff and I'd not let my birds inhale it so soon afterwards for sure. That will kill red mite stone dead and prevent any reinfestation anytime this year.
I also find Dettol to clean the hen house out works well each week then as does encouraging earwigs to take up residence in all corners of your hen house (making sure that when you clean the house out you safely remove the earwigs and stash them in a jam jar and then put them back in the hen house once its clean and dry)
I'd be looking at other possibilities if its not red mite. As in infection spreading amongst your flock. Did you quarantine your new birds for a while before adding them in with your current birds? How are their droppings looking? Perhaps take a sample of droppings to your vet for analysis (and bloods also?) if you've ruled out red mite.
