The biblical prohibition against mixing meat and milk (Shemot 23:19) is extended by the Mishnah to forbid cooking any meat with milk and even placing meat and cheese together on the same table (Mishnah Chullin 8:1), forming the foundation for all subsequent discussion about shared vessels.
The operative halachic principle for washing mixed vessels is noten ta'am bar noten ta'am — a secondary imparted flavor — which the Gemara traces to the dispute between Rav and Shmuel regarding fish placed in a meat bowl: Rav prohibits eating the fish with a dairy dish because flavor has been directly imparted, while Shmuel permits it as only a secondary imparted flavor (Chullin 111b), and the Arukh HaShulchan (Yoreh De'ah 95) clarifies that this leniency of secondary flavor applies specifically to the meat-milk combination, whereas with a prohibited substance the chain of transmission never diminishes.
The Shulchan Arukh (Yoreh De'ah 95:3) rules that meat dishes washed in a dairy boiler filled with scalding water are permitted even if both are ben yomo, on account of secondary imparted flavor, provided there is no fatty residue on the dishes — but the Rema there records the stricter Ashkenazic practice that if both vessels are ben yomo and were washed together in a kli rishon, everything is forbidden, and this practice is not to be departed from.
The Beit Yosef explains the stricter position by noting that when meat and dairy flavors enter the water simultaneously, the water itself becomes forbidden immediately and then re-prohibits all the vessels in contact with it (Beit Yosef, Yoreh De'ah 95), a reasoning that bears directly on whether a single dishwasher may be used for both types of vessels.