  We prove stable versions of trace theorems on the sphere in $L^2$ with
optimal constants, thus obtaining rather precise information regarding
near-extremisers. We also obtain stability for the trace theorem into $L^q$ for
$q > 2$, by combining a refined Hardy-Littlewood-Sobolev inequality on the
sphere with a duality-stability result proved very recently by Carlen. Finally,
we extend a local version of Carlen's duality theorem to establish local
stability of certain Strichartz estimates for the kinetic transport equation.
