I am interested in the following question, which is stated in the abstract of [1]:

An urn contains $r$ different balls. Balls are drawn with replacement until any $k$ balls have been obtained at least $m$ times each. How many draws are necessary?

The cited paper solves this problem using Poissonization for various special cases (e.g. $k=1$ or $k=r$ and so on) but I cannot find a reference for the most general case, asymptotic or otherwise. Is there a name for this general problem, and are there any known solutions (or lower bounds) for it, assuming that $r\to\infty$?

[1] Holst, Lars. "On birthday, collectors', occupancy and other classical urn problems." International Statistical Review/Revue Internationale de Statistique (1986): 15-27.