Finitely dependent processes generalize independent processes by requiring that the restrictions of the process to sufficiently separated sets are independent. The existence of stationary finitely dependent processes on combinatorial models like $\mathbb Z^d$ subshifts can be quite mysterious. For instance, Holroyd and Liggett constructed such processes on proper $4$-colorings of $\mathbb Z^d$ for all $d$ while Holroyd, Schramm and Wilson showed that there are no such processes on proper $3$-colorings of $\mathbb Z^d$ for $d>1$. In this paper, we take inspiration from these results and investigate them further. On the positive side, we show that there exists a dense set of stationary finitely dependent processes supported on subshifts with strong mixing properties like the finite extension property. On the negative side, we see that the cohomology of the subshifts can form an obstruction to the existence of such processes. In particular we use Conway-Lagarias-Thurston height functions to characterize when there exists a finitely dependent process on the space of tilings by boxes of $\mathbb Z^2$ answering the tiling problem posed by Gao, Jackson, Krohne and Seward in dimension $2$. The ideas also apply to many other models, such as graph homomorphisms and ribbon tilings. On the way, we also show that continuous cocycles on strongly irreducible subshifts valued in a special class of groups (including torsion free Gromov hyperbolic groups and free product of cyclic groups) are perturbations of group homomorphisms.