  In this paper, we derive analytical expressions for the leading-order
hydrodynamic mobility of a small solid particle undergoing motion tangential to
a nearby large spherical capsule whose membrane possesses resistance towards
shearing and bending. Together with the results obtained in the first part
(Daddi-Moussa-Ider and Gekle, Phys. Rev. E {\bfseries 95}, 013108 (2017)) where
the axisymmetric motion perpendicular to the capsule membrane is considered,
the solution of the general mobility problem is thus determined. We find that
shearing resistance induces a low-frequency peak in the particle self-mobility,
resulting from the membrane normal displacement in the same way, although less
pronounced, to what has been observed for the axisymmetric motion. In the zero
frequency limit, the self-mobility correction near a hard sphere is recovered
only if the membrane has a non-vanishing resistance towards shearing. We
further compute the particle in-plane mean-square displacement of a nearby
diffusing particle, finding that the membrane induces a long-lasting
subdiffusive regime. Considering capsule motion, we find that the correction to
the pair-mobility function is solely determined by membrane shearing
properties. Our analytical calculations are compared and validated with fully
resolved boundary integral simulations where a very good agreement is obtained.
