For $t \in \mathbb{N}$ and every $i\in[t]$, let $H_i$ be a $d_i$-regular connected graph, with $1<|V(H_i)|\le C$ for some integer $C\ge 2$. Let $G=\square_{i=1}^tH_i$ be the Cartesian product of $H_1, \ldots, H_t$. We show that if $t\ge 5C$ then $G$ contains a (nearly-)perfect matching. Then, considering the random graph process on $G$, we generalise the result of Bollobás on the binary hypercube $Q^t$, showing that with high probability, the hitting times for minimum degree one, connectivity, and the existence of a (nearly-)perfect matching in the random graph process on $G$ are the same. As a byproduct, we develop several tools which may be of independent interest in a more general setting when one seeks to establish the typical existence of a perfect matching under percolation.