Matrix products of binomial coefficients and unsigned Stirling numbers

We study sums of the form ${\sum }_{k=m}^n{a}_{nk}{b}_{km}$, where $a_{nk}$ and $b_{km}$ are binomial coefficients or unsigned Stirling numbers. In a few cases they can be written in closed form. Failing that, the sums still share many common features: combinatorial interpretations, Pascal-like recurrences, inverse relations with their signed versions, and interpretations as coefficients of change between polynomial bases.