In the setting you are interested in, that is, finitely generated k-algebras and GIT-quotients of closed or semistable orbits, Morita equivalence induces isomorphism on the moduli spaces provided one scales dimension(vectors) and stability structures accordingly. That is, if B=M_n(A) one should compare Mod(A,k) to Mod(B,nk) moduli. If A and B have a complete set of orthogonal idempotents e_i resp. f_i(that is, a quiver-like situation) and if the Morita equivalence induces rank(f_i) = n_i rank(e_i), then one should compare A-reps of dimension vector alpha=(alpha_i) to B-reps with dimension vector beta=(n_i alpha_i).

Geometrically, the module varieties of Morita-equivalent algebras are related via associated fibre bundle constructions. In the example above, Mod(B,kn) = GL(nk) x^GL(k) Mod(A,k). In general,one had such a description locally in the Zariski topology (coming roughly from the fact that a vectorbundle (or projective) is locally free). Anyway, this gives a natural and geometric one-to-one correspondence between GL(nk)-orbits (isoclasses) of B-reps and GL(k)-orbits of A-reps inducing the desired isos on the quotient variety level (isos of semi-simples). In quiver-like situations one should adjust the dim vectors as above.

Now as to semi-stability. Observe that semi-stable reps are just ordinary reps of a universal localization of your algebra(s), so one can reduce to the closed case by covering the variety of semi-stables by Zariski open (affine) pieces. In quiver-like situations when the Morita-data is as above and your stability structure mu for A is given by the vector (mu_i), then the corresponding stability structure for B is mu'=(1/n_i x mu_i) (or multiply it with a common factor if you want it to have integral components.