We consider  parabolic equations of the form
\begin{equation*}
u_t = \Delta u + f(u) + h(x, t), \quad (x, t) \in R^N \times (0,\infty)\,,
\end{equation*}
where  $f$ is a $ C^1$ function  with $f(0) = 0$, $f'(0) < 0$, and $h$ is a suitable function  on $R^N\times [0,\infty)$ which decays to zero as $t\to\infty$ (hence the equation is asymptotically autonomous).  We show that, as $t\to \infty$, each bounded localized solution $u\ge 0$ approaches a set of steady states of the limit  autonomous equation $ u_t = \Delta u + f(u)$. Moreover, if the decay of $h$ is exponential, then $u$ converges to a single steady state. We also prove a convergence result for abstract asymptotically autonomous parabolic equations.