If we have a quantum channel mapping from a $d$-dimensional state to a $d$-dimensional state, it can be described by at most $d^2$ Kraus operators. Suppose our channel maps instead from a $d_1$-dimensional state to a $d_2$-dimensional state, with $d_1>d_2$, e.g. with the quantum operation of taking the partial trace over a mode. What is the maximum required number of Kraus operators to characterise the channel? Is it $d_1d_2$, analogous to the case where $d_1=d_2$?
Yes. Choi's Theorem a priori uses different Hilbert spaces of potentially different dimensions $d_1$ and $d_2$. Then $d_1=d_2$ is a corollory. The proof is included there.