  The influence of the chiral mean field on the $K^+$ transverse flow in heavy
ion collisions at SIS energy is investigated within covariant kaon dynamics.
For the kaon mesons inside the nuclear medium a quasi-particle picture
including scalar and vector fields is adopted and compared to the standard
treatment with a static potential. It is confirmed that a Lorentz force from
spatial component of the vector field provides an important contribution to the
in-medium kaon dynamics and strongly counterbalances the influence of the
vector potential on the $K^+$ in-plane flow. The FOPI data can be reasonably
described using in-medium kaon potentials based on effective chiral models. The
information on the in-medium $K^+$ potential extracted from kaon flow is
consistent with the knowledge from other sources.
