**Abstract** : The slow viscous and either imposed or gravity-driven migration of a solid arbitrarily-shaped particle suspended in a Newtonian liquid bounded by a spherical cavity is calculated using two different boundary element approaches. Each advocated method appeals to a few boundary-integral equations and, by contrast with previous works, also holds for non-spherical particles. The first procedure puts usual free-space Stokeslets on both the cavity and particle surfaces whilst the second one solely spreads specific Stokeslets obtained elsewhere in Oseen (1927) on the particle's boundary. Each approach receives a numerical implementation which is found to be in excellent agreement with accurate results available for spherical particles. The computations for spheroidal or ellipsoidal particles, here accurately achieved at a very reasonable cpu time cost using the second technique, reveal that the particle settling migration deeply depends upon the gravity and upon both its shape and location inside the cavity.