Ken Brown shows in 

* _Homological criteria for finiteness_, Comment. Math. Helv. 50 (1975), 129–135, doi:[10.1007/BF02565740](https://doi.org/10.1007/BF02565740), ([free author version](https://pi.math.cornell.edu/~kbrown/scan/1975.0050.pdf))

that group cohomology for a group G commutes with direct limits iff G is of type $FP_\infty$. That is the trivial module $\mathbb Z$ has a projective $\mathbb ZG$ resolution which is finitely generated in each degree.