  We present a posteriori error estimates for inconsistent and non-hierarchical
Galerkin methods for linear parabolic problems, allowing them to be used in
conjunction with very general mesh modification for the first time. We treat
schemes which are non-hierarchical in the sense that the spatial Galerkin
spaces used on consecutive time-steps may be completely unrelated from one
another. The practical interest of this setting is demonstrated by applying our
results to finite element methods on moving meshes and using the estimators to
drive an adaptive algorithm based on a virtual element method on a mesh of
arbitrary polygons. The a posteriori error estimates, for the error measured in
the $L^2(H^1)$ and $L^{\infty}(L^2)$ norms, are derived using the elliptic
reconstruction technique in an abstract framework designed to precisely
encapsulate our notion of inconsistency and non-hierarchicality and requiring
no particular compatibility between the computational meshes used on
consecutive time-steps, thereby significantly relaxing this basic assumption
underlying previous estimates.
