Convergence in Mallows distance is of particular interest when heavy-tailed distributions are considered. For 1≤α<2, it constitutes an alternative technique to derive Central Limit type theorems for non-Gaussian α-stable laws. In this note, for properly stabilized martingale sums and sequences of φ-mixing random variables, we establish Mallows convergence to stable laws. Sufficient conditions are presented in the setting of familiar Lindeberg-like conditions and extend earlier results for the independent case.