<p style='text-indent:20px;'>In this work we prove continuity of solutions with respect to initial conditions and a couple of parameters and we prove upper semicontinuity of a family of pullback attractors for the problem <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$\left\{ {\begin{array}{*{20}{l}}{\frac{{\partial {u_s}}}{{\partial t}}(t) - {D_s}{\rm{div}}(|\nabla {u_s}{|^{{p_s}(x) - 2}}\nabla {u_s}) + C(t)|{u_s}{|^{{p_s}(x) - 2}}{u_s} = B({u_s}(t)),\;\;t > \tau ,}\\{{u_s}(\tau ) = {u_{\tau s}},}\end{array}} \right.$ \end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>under homogeneous Neumann boundary conditions, <inline-formula><tex-math id="M1">\begin{document}$u_{τ s}∈ H: = L^2(Ω),$\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M2">\begin{document}$Ω\subset\mathbb{R}^n$\end{document}</tex-math></inline-formula>(<inline-formula><tex-math id="M3">\begin{document}$n≥ 1$\end{document}</tex-math></inline-formula>) is a smooth bounded domain, <inline-formula><tex-math id="M4">\begin{document}$B:H \to H$\end{document}</tex-math></inline-formula> is a globally Lipschitz map with Lipschitz constant <inline-formula><tex-math id="M5">\begin{document}$L≥ 0$\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$D_s∈[1,∞)$\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$C(·)∈ L^{∞}([τ, T];\mathbb{R}^+)$\end{document}</tex-math></inline-formula> is bounded from above and below and is monotonically nonincreasing in time, <inline-formula><tex-math id="M8">\begin{document}$p_s(·)∈ C(\bar{Ω})$\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M9">\begin{document}$p_s^-: = \textrm{min}_{x∈\bar{Ω}}\;p_s(x)≥ p,$\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M10">\begin{document}$p_s^+: = \textrm{max}_{x∈\bar{Ω}}\;p_s(x)≤ a,$\end{document}</tex-math></inline-formula> for all <inline-formula><tex-math id="M11">\begin{document}$s∈ \mathbb{N},$\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M12">\begin{document}$p_s(·) \to p$\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M13">\begin{document}$L^∞(Ω)$\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M14">\begin{document}$D_s \to ∞$\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M15">\begin{document}$s \to∞,$\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M16">\begin{document}$a,p>2$\end{document}</tex-math></inline-formula> positive constants.