  We introduce multinomial and $r$-variants of several classic objects of
combinatorial probability, such as the random recursive and Hoppe trees, random
set partitions and compositions, the Chinese restaurant process, Feller's
coupling, and some others. Just as various classic combinatorial numbers - like
Stirling, Eulerian and Lah numbers - emerge as essential ingredients defining
the distributions of the mentioned processes, the so-called $r$-versions of
these numbers appear in exact distributional formulas for the multinomial and
$r$-counterparts. This approach allows us to offer a concise probabilistic
interpretation for various identities involving $r$-versions of these
combinatorial numbers, which were either unavailable or meaningful only for
specific values of the parameter $r$. We analyze the derived distributions for
fixed-size structures and establish distributional limit theorems as the size
tends to infinity. Utilizing the aforementioned generalized Stirling numbers of
both kinds, we define and analyze $(r,s)$-Lah distributions, which have arisen
in the existing literature on combinatorial probability in various contexts.
