It is established that if

$X$ is a Banach space, then the composition operator on

$X$-valued Hardy spaces, weighted Bergman spaces, and on Bloch spaces is weakly compact, respectively Rosenthal (i.e., strictly cosingular) if and only if both the identity operator on

$X$ and the corresponding composition operator on the analogous scalar valued spaces have this same property. The final section of the paper contains some results for ‘general vector-valued spaces’, where the range space is defined in terms of a Banach space of analytic functions on the unit disc whose closed unit ball is compact in the compact open topology.