The foundational prohibition begins at Shemot 23:19, which forbids cooking a kid in its mother's milk, and Mishnah Chullin 8:1 extends that prohibition to all meat, barring it from being cooked with milk or even brought to the table alongside cheese.
The Gemara in Chullin 111b establishes the principle of נותן טעם בר נותן טעם — imparted flavor derived from imparted flavor — ruling that hot fish placed in a meat bowl may still be eaten with a dairy dish, because the meat flavor passed first into the bowl and only then into the fish, attenuating it sufficiently; Arukh HaShulchan, Yoreh De'ah 95 explains that this leniency applies only to the meat-dairy combination, where the twice-transferred flavor is so weakened that it does not create a prohibition, whereas with other forbidden substances the flavor retains its force no matter how many steps removed.
Shulchan Arukh, Yoreh De'ah 95:3 codifies the case of meat dishes washed in a dairy boiler full of hot water: where no fatty residue clings to the dishes, the result is permitted as נותן טעם בר נותן טעם, but the Rema's gloss records the stringent Ashkenazic practice that if both vessels are ben-yomo and were washed together in a kli rishon, everything is forbidden — and this is the normative custom not to be changed.
Iggerot Moshe, Yoreh Deah 2:28 concludes, on our reading of the cited page, that the racks or compartments inside a dishwasher require separation — one for meat vessels and one for dairy vessels — referencing his earlier ruling that one must in effect maintain two distinct dishwasher units or at minimum two separate racks, one for meat and one for dairy.