Consider random sequential adsorption on a chequerboard lattice with arrivals at rate 1 on light squares and at rate λ on dark squares. Ultimately, each square is either occupied, or blocked by an occupied neighbour. Colour the occupied dark squares and blocked light sites black, and the remaining squares white. Independently at each meeting-point of four squares, allow diagonal connections between black squares with probability p; otherwise allow diagonal connections between white squares. We show that there is a critical surface of pairs (λ, p), containing the pair (1,0.5), such that for (λ, p) lying above (respectively, below) the critical surface the black (resp. white) phase percolates, and on the critical surface neither phase percolates. We find conditions satisfied by a broad class of essentially planar percolation models such that for a model satisfying the conditions, the presence or absence of percolation is determined by what happens in a collection of finite boxes. This criterion applies to a (non-degenerate) Poisson Boolean model, to the random connection model for some sufficiently high p < 1, and to the model described above. We also find conditions that do not require rotation invariance which produce a comparable result; these conditions seem plausible, but finding non-trivial examples is a matter for further research.