Summary: We propose and study a reaction-diffusion model with memory, where the linear diffusion operator is replaced by the mean curvature operator both in Euclidean and Lorentz-Minkowski spaces and the memory kernel is assumed to be of Jeffreys type. Regarding the reaction term, we consider a balanced bistable function \(f = - F^\prime\), with \(F\) a generic double well potential with wells of equal depth. In particular, we assume that the potential \(F\) has two global minima at \(\pm 1\) and that \(F(u) \sim | 1 \pm u |^{2 + \theta}\), for some \(\theta > - 1\), when \(u \approx \pm 1\), and we consider the corresponding equation in a bounded interval with homogeneous Neumann boundary conditions. We prove that if \(\theta \in(- 1, 0)\), then there exist special steady states, named compactons, with a transition layer structure. In contrast, if \(\theta \geq 0\) the interface layers are not stationary and two different phenomena emerge: for \(\theta = 0\) solutions exhibit a metastable behavior and maintain an unstable structure for an exponentially long time, while if \(\theta > 0\) the exponentially slow motion is replaced by an algebraic one.