Drawing colored balls without replacement; tracking maximum balls picked across different colors

Question

The following problem is a part of a research problem in a different field that I am working on. I have removed the details from the field on purpose because it is far removed from this piece as such but this is a crucial part of it. So far I have looked into multivariate hypergeometric distribution and the literature on balls and bins problems. I have not encountered this variant or anything that tracks the maximum occupancy in this setting instead of average occupancy across different colors.

Say we are given an urn with m₁ balls of color C₁, m₂ balls of color C₂, .., m_k balls of color C_k such that

m₁ + m₂ + m₃ + .. + m_k = m.

Suppose we draw l <= m balls from the urn without replacement. Let the number of balls drawn of each color be l₁, l₂,..,l_k such that

l₁ + l₂ + .. + l_k = l.

What is the probability that max(l₁,l₂,..l_k) = t?

If this case is too general and non-tractable, I would also be happy to know leads for the following special cases:

(a) m₁=m₂=..=m_k,

and/or

(b) k = 4.

Any leads will be greatly appreciated. Thanks!