  Because different constraints are imposed, stability conditions for
dissipationless fluids and magnetofluids may take different forms when derived
within the Lagrangian, Eulerian (energy-Casimir), or dynamical accessible
frameworks. This is in particular the case when flows are present. These
differences are explored explicitly by working out in detail two
magnetohydrodynamic examples: convection against gravity in a stratified fluid
and translationally invariant perturbations of a rotating magnetized plasma
pinch. In this second example we show in explicit form how to perform the
time-dependent relabeling introduced in Andreussi {\it et al.}\ [Phys.\ Plasmas
{\bf20}, 092104 (2013)] that makes it possible to reformulate Eulerian
equilibria with flows as Lagrangian equilibria in the relabeled variables. The
procedures detailed in the present article provide a paradigm that can be
applied to more general plasma configurations and in addition extended to more
general plasma descriptions where dissipation is absent.
