We isolate a general condition, that we call "localization principle", on the integrand $L:\mathbb{M}\to[0,\infty]$, assumed to be continuous, under which $W^{1,q}$-quasiconvexity with $q\in[1,\infty]$ is a sufficient condition for $I(u)=\int_\Omega L(\nabla u(x))dx$ to be sequentially weakly lower semicontinuous on $W^{1,p}(\Omega;\mathbb{R}^m)$ with $p\in]1,\infty[$. Some applications are given.