  Given a non-empty closed convex subset $\mathsf{F}$ of density matrices, we
formulate conditions that guarantee the existence of an $\mathsf{F}$-morphism
(namely, a completely positive trace-preserving linear map that maps
$\mathsf{F}$ into itself) between two arbitrarily chosen density matrices.
While we allow errors in the transition, the corresponding map is required to
be an exact $\mathsf{F}$-morphism. Our findings, though purely geometrical, are
formulated in a resource-theoretic language and provide a common framework that
comprises various resource theories, including the resource theories of
bipartite and multipartite entanglement, coherence, athermality, and asymmetric
distinguishability. We show how, when specialized to some situations of
physical interest, our general results are able to unify and extend previous
analyses. We also study conditions for the existence of maximally resourceful
states, defined here as density matrices from which any other one can be
obtained by means of a suitable $\mathsf{F}$-morphism. Moreover, we
quantitatively characterize the paradigmatic tasks of optimal resource dilution
and distillation, as special transitions in which one of the two endpoints is
maximally resourceful.
