  In this paper, we investigate the continuity of the attractors in
time-dependent phase spaces. (i) We establish two abstract criteria on the
upper semicontinuity and the residual continuity of the pullback $\mathscr
D$-attractor with respect to the perturbations, and an equivalence criterion
between their continuity and the pullback equi-attraction, which generalize the
continuity theory of attractors developed recently in [27,28] to that in
time-dependent spaces. (ii) We propose the notion of pullback $\mathscr
D$-exponential attractor, which includes the notion of time-dependent
exponential attractor [33] as its spacial case, and establish its existence and
H\"{o}lder continuity criterion via quasi-stability method introduced
originally by Chueshov and Lasiecka [12,13]. (iii) We apply above-mentioned
criteria to the semilinear damped wave equations with perturbed time-dependent
speed of propagation: $\e\rho(t) u_{tt}+\alpha u_t -\Delta u+f(u)=g$, with
perturbation parameter $\e\in(0, 1]$, to realize above mentioned continuity of
pullback $\mathscr D$ and $\mathscr D$-exponential attractors in time-dependent
phase spaces, and the method developed here allows to overcome the difficulty
of the hyperbolicity of the model. These results deepen and extend recent
theory of attractors in time-dependent spaces in literatures [15,20,19].
