  We study the dynamical process of braiding Majorana bound states in the
presence of the coupling to photons in a microwave cavity. We show
theoretically that the $\pi/4$ phase associated with the braiding of Majoranas,
as well as the parity of the ground state are imprinted into the photonic field
of the cavity, which can be detected by dispersive readouts techniques. These
manifestations are purely dynamical, they occur in the absence of any splitting
of the MBS that are exchanged, and they disappear in the static setups studied
previously. Conversely, the cavity can affect the braiding phase, which in turn
should allow for cavity controlled braiding.
