  The stability of a collisionless, magnetised plasma to local convective
disturbances is examined, with a focus on kinetic and finite-Larmor-radius
effects. Specific application is made to the outskirts of galaxy clusters,
which contain hot and tenuous plasma whose temperature increases in the
direction of gravity. At long wavelengths (the "drift-kinetic" limit), we
obtain the kinetic version of the magnetothermal instability (MTI) and its
Alfv\'enic counterpart (Alfv\'enic MTI), which were previously discovered and
analysed using a magnetofluid (i.e. Braginskii) description. At sub-ion-Larmor
scales, we discover an overstability driven by the electron temperature
gradient of kinetic-Alfv\'en drift waves -- the electron MTI (eMTI) -- whose
growth rate is even larger than the standard MTI. At intermediate scales, we
find that ion finite-Larmor-radius effects tend to stabilise the plasma. We
discuss the physical interpretation of these instabilities in detail, and
compare them both with previous work on magnetised convection in a collisional
plasma and with temperature-gradient-driven drift-wave instabilities well-known
to the magnetic-confinement-fusion community. The implications of having both
fluid and kinetic scales simultaneously driven unstable by the same temperature
gradient are briefly discussed.
