We prove that for any field $k$ of characteristic $p{>}0$ , any separated scheme $X$ of finite type over $k$ , and any overconvergent $F$ -isocrystal ${\cal E}$ over $X$ , the rigid cohomology $H^i_{\rm rig}(X, {\cal E})$ and rigid cohomology with compact supports $H^i_{c,{\rm rig}}(X, {\cal E})$ are finite-dimensional vector spaces over an appropriate $p$ -adic field. We also establish Poincaré duality and the Künneth formula with coefficients. The arguments use a pushforward construction in relative dimension $1$ , based on a relative version of Crew's [Cr] conjecture on the quasi-unipotence of certain $p$ -adic differential equations