If $\vf_1, ... \vf_m\colon\Z\to\Z^\ell$ are polynomials with zero constant
terms and $E\subset\Z^\ell$ has positive upper Banach density, then we show
that the set $E\cap (E-\vf_1(p-1))\cap\...\cap (E-\vf_m(p-1))$ is nonempty for
some prime $p$. We also prove mean convergence for the associated averages
along the prime numbers, conditional to analogous convergence results along the
full integers. This generalizes earlier results of the authors, of Wooley and
Ziegler, and of Bergelson, Leibman and Ziegler.