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.