I have a problem in my new kitchen. After eating meat, I put my meat dishes in the meat sink and hand wash them using hot water and detergent, then rinse them with hot water in the meat sink, then put them in a plate rack to air-dry. While they’re drying, drops of water may drip off the plates. Suppose they land in the milk sink? Does that matter? Once the plate has been washed, can drops of the hot/warm rinsing water carry meatiness from meat plates onto milk plates?

Not a problem at all.