By means of Monte Carlo computer simulations we study the domain-growth kinetics after a quench across a first-order line to very low and moderate temperatures in a multidegenerate system with nonconserved order parameter. The model is a continuous spin model relevant for martensitic transformations, surface reconstructions, and magnetic transitions. No external impurities are introduced, but the model has a number of intrinsic, annealable pinning mechanisms, which strongly influences the growth kinetics. It allows a study of pinning effects of three kinds: (a) pinning of domain walls by defects-this is found in effect to stop the growth, forming a metastable state at low temperatures T; (b) temporary pinning by stacking faults or zero-curvature domain walls; and (c) topological pinnings, which are also found to be temporary. These just slow down the growth. The pinning mechanisms and the depinning probability at higher temperatures are studied. The excess energy of the domain walls is found to follow an algebraic decay DELTA-E(t) = E(M) + At(-n), with E(M) = 0 for cases (b) and (c) and decaying toward a metastable state with energy E(M) not-equal-to 0 for case (a). The exponent is found to cross over from n = 1/4 at T approximately 0 to n = 1/2 with temperature for models with pinnings of types (a) and (b). For topological pinnings at T approximately 0, n is consistent with n = 1/8, a value conceivable for several levels of hierarchically interrelated domain-wall movement. When the continuous-spin model is reduced to a discrete Potts-like model, with the same parameters, the exponent is found to be consistent with the classical Allen-Cahn exponent n = 1/2.