In this talk, we consider partial differential equations of parabolic $p$-Laplace type in noncylindrical domains $U = \bigcup_{t \in [0,T]} \{t\} \times \Omega_t$, where the underlying spatial set $\Omega_t \subset \mathbb{R}^n$ may change in time. Our goal is to characterize time-dependent boundary values that yield the existence of a solution, focusing on domains $U$ that satisfy a Lipschitz condition with respect to a parabolic metric. In particular, we discuss the extension to $U$ of certain functions that are only defined on the parabolic boundary of $U$, and conversely the trace space for functions defined on $U$ and contained in a suitable parabolic function space. The talk is based on joint work with Daniel Campbell and Sebastian Schwarzacher.