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The present paper deals with the asymptotic behavior of equi-coercive sequences $\{\mathcal{F}_n\}$ of nonlinear functionals defined over vector-valued functions in $W_)^{1,p}(\Omega)^M$ , where $p>1$, $M\ge1$, and $\Omega$ is a bounded open set of $\mathbb{R}^N$, $N\ge2$. The strongly local energy density $F_n({\cdot}, Du)$ of the functional $\{\mathcal{F}_n\}$ satisfies a Lipschitz condition with respect to the second variable, which is controlled by a positive sequence $\{a_n\}$ which is only bounded in some suitable space $L^r(\Omega)$. We prove that the sequence $\{\mathcal{F}_n\}$ $\Gamma$-converges for the strong topology of $L^p(\Omega)^M$ to a functional $\mathcal{F}$ which has a strongly local density $F({\cdot}, Du)$ for sufficiently regular functions $u$. This compactness result extends former results on the topic, which are based either on maximum principle arguments in the nonlinear scalar case, or adapted div-curl lemmas in the linear case. Here, the vectorial character and the nonlinearity of the problem need a new approach based on a careful analysis of the asymptotic minimizers associated with the functional $\mathcal{F}_n$. The relevance of the conditions which are imposed to the energy density $F_n({\cdot}, Du)$, is illustrated by several examples including some classical hyper-elastic energies.