They aren't red mite.
They are a sort of dust mite, and whereas red mite tend to congregate in the areas where poultry roost, these will be found anywhere in a building. Like you say, they appear to drop on you from above, as they get onto spiders' webs which touch your hair as you pass.
You can get rid of them in much the same way as red mites, although they are not so persistent. So a good spray round with insecticide, or creosote diluted with paraffin/diesel usually sorts them.
They are quite easily brushed out of your hair or off your clothing, preferably before you enter the house. They don't live on furniture, but you may find a few escapees still crawling over you in the evening. It's annoying but you can just pick them off and squash them between your fingers.
Once you've had a spray round you'll find things a lot better! But in the meantime, don't touch anything you can avoid in the poultry shed, and if you do, then look at your hands or whatever, and brush the creatures off before they get all over you.