We have cups for meat and milk that are plastic for the kids. We never put milk in the meat cups, however, it has happened that a meat cup with water has sat out and the next meal (milk) was set on the table. If they drink from the cup is there a problem?
Very good question. Since the cup was used to drink during the met meal, we can not use it during a milk meal unless it is washed out first to get rid of any leftover residue etc. from the other meal.
Regarding drinking from the cup after eating a milk meal, then it also should preferably be washed out, however then it is only a preference.
Y:D 88-2, although the above is talking about two people eating at the same time, it would apply also to one person eating meat while drinking from the cup, and then drinking from the cup while he eats milk. however this is only when they are eating together, but afterwards is different, see Badei Hashulchan 88-28, Zerah Shmuel Ftnt 92, Ohel Yackov 89-18.