Summary: Let \(G_n\) be a sequence of open subsets of a given open and bounded \(\Omega\subset \mathbb{R}^N\). We study the asymptotic behaviour of the solutions of parabolic equations \(u_n'+Au_n=f_n\) on \(G_n\). Assuming that the right-hand sides \(f_n\) and the initial conditions converge in a proper way we find the form of the limit problem without any additional hypothesis on \(G_n\). Our method is based on the notion of elliptic \(\gamma^A\)-convergence.