Poultry Shield has the same effect on Red Mite as washing up liquid. It hasn't been sold as a Red Mite killer for years -so you must have an old bottle. Smite is good at flushing out eggs as it contains a powerful detergent that 'unsticks' them, but it doesn't kill red mite very quickly. Nettex Total Mite Kill does -but doesn't wash out the eggs. The aformentioned two work well in a combination of Smite followed immediately by Nettex.
Alternatives. Steamers work well with a fine nozzle (particularly into the weather boarding joints)and we usually start with that and follow up with chemicals. Blow torch has much same effect but risks burning down the coop. Jetwasher is good but can split or shred the wood, but it washes out the preservatives as well and just moves the mite and eggs with the water. Plastic coop like Green Frog is very easy to strip and clean completely. Creosote, the ultimate red mite killer, which may shortly be banned. Creocote does work but not quite as well. Mite love new untreated wood so just get something into it. Or the radical -Elector a biocide, which targets the nerve system of the mite. Really easy to apply as you just spray everything in sight. Totally effective for a few months at least but the mites evolve immunity so can't use it two years in a row. Has a very long shelf life though -5 years I think. Woodash soil baths -recommended in France for control on the birds and it seems to work. Haven't tried anything else as I am still using my 30 litre drum of creosote (diluted 50:50 with paraffin for better application).