An extension of the pioneer work by Mullins and Sekerka is proposed to analyze the stability conditions of a spherical particle growing in a matrix phase. The present model describes the onset of instabilities in an unsteady growth regime for multi-component alloys with cross-diffusion of chemical species. The developments are based on the decomposition of initial perturbations in spherical harmonics to determine their time evolution. Expressions for the threshold particle radii associated to the limits for absolute and relative stability criteria are derived. They depend on the degree of the spherical harmonic functions, the growth parameter for a sphere (proportional to the interface velocity times the square root of time) and the eigenvalues of the diffusion matrix. A large effect of the interface velocity on the stability domains is demonstrated considering the time-dependent solutions. Comparisons are provided regarding more recent solutions proposed in the literature. A similar evolution is only observed for slow regimes. An application for solidification of a ternary alloy is finally given.