Although all-in-one-model multilingual neural machine translation (MNMT) has achieved remarkable progress, the convergence inconsistency in the joint training is ignored, i.e.,different language pairs reaching convergence in different epochs. This leads to the trained MNMT model over-fitting low-resource language translations while under-fitting high-resource ones. In this paper, we propose a novel training strategy named LSSD (LanguageSpecific Self-Distillation), which can alleviate the convergence inconsistency and help MNMT models achieve the best performance on each language pair simultaneously. Specifically, LSSD picks up language-specific best checkpoints for each language pair to teach the current model on the fly. Furthermore, we systematically explore three sample-level manipulations of knowledge transferring. Experimental results on three datasets show that LSSD obtains consistent improvements towards all language pairs and achieves the state-of-the-art.