In this work we study the continuity of four different notions of asymptotic behavior for a family of non-autonomous non-classical parabolic equations given by

{ut−γ(t)Δut−Δu=gϵ(t,u), in Ωu=0, on ∂Ω.
$$\begin{array}{}
\displaystyle
\left\{ \begin{array}{*{20}{l}}
{{u_t} - \gamma \left( t \right)\Delta {u_t} - \Delta u = {g_\varepsilon }\left( {t,u} \right),{\;\text{in}\;}\Omega } \hfill \\
{u = 0,{\;\text{on}\;}\partial \Omega {\rm{.}}} \hfill \\
\end{array}\right.
\end{array}$$
in a smooth bounded domain Ω ⊂ ℝn, n ⩾ 3, where the terms gε are a small perturbation, in some sense, of a function f that depends only on u.