  Bootstrap percolation on a graph is a deterministic process that iteratively
enlarges a set of occupied sites by adjoining points with at least $\theta$
occupied neighbors. The initially occupied set is random, given by a uniform
product measure with a low density $p$. Our main focus is on this process on
the product graph $\mathbb{Z}^2\times K_n^2$, where $K_n$ is a complete graph.
We investigate how $p$ scales with $n$ so that a typical site is eventually
occupied. Under critical scaling, the dynamics with even $\theta$ exhibits a
sharp phase transition, while odd $\theta$ yields a gradual percolation
transition. We also establish a gradual transition for bootstrap percolation on
$\mathbb{Z}^2\times K_n$. The main tool is heterogeneous bootstrap percolation
on $\mathbb{Z}^2$.
