  We present a study of in-medium cross sections and (off-shell) transition
rates for the most relevant binary reactions for strange pseudoscalar meson
production close to threshold in heavy-ion collisions at FAIR energies. Our
results rely on a chiral unitary approach in coupled channels which
incorporates the $s$- and $p$-waves of the kaon-nucleon interaction. The
formalism, which is modified in the hot and dense medium to account for Pauli
blocking effects, mean-field binding on baryons, and pion and kaon
self-energies, has been improved to implement full unitarization and
self-consistency for both the $s$- and $p$-wave interactions at finite
temperature and density. This gives access to in-medium amplitudes in several
elastic and inelastic coupled channels with strangeness content $S=-1$. The
obtained total cross sections mostly reflect the fate of the $\Lambda(1405)$
resonance, which melts in the nuclear environment, whereas the off-shell
transition probabilities are also sensitive to the in-medium properties of the
hyperons excited in the $p$-wave amplitudes [$\Lambda$, $\Sigma$ and
$\Sigma^*(1385)$]. The single-particle potentials of these hyperons at finite
momentum, density and temperature are also discussed in connection with the
pertinent scattering amplitudes. Our results are the basis for future
implementations in microscopic transport approaches accounting for off-shell
dynamics of strangeness production in nucleus-nucleus collisions.
