  In this work we study the stability, convergence, and pressure-robustness of
discretization methods for incompressible flows with hybrid velocity and
pressure. Specifically, focusing on the Stokes problem, we identify a set of
assumptions that yield inf-sup stability as well as error estimates which
distinguish the velocity- and pressure-related contributions to the error. We
additionally identify the key properties under which the pressure-related
contributions vanish in the estimate of the velocity, thus leading to
pressure-robustness. Several examples of existing and new schemes that fit into
the framework are provided, and extensive numerical validation of the
theoretical properties is provided.
