We propose a practical scheme to train a single multilingual sequence labeling model that yields state of the art results and is small and fast enough to run on a single CPU. Starting from a public multilingual BERT checkpoint, our final model is 6x smaller and 27x faster, and has higher accuracy than a state-of-the-art multilingual baseline. We show that our model especially outperforms on low-resource languages, and works on codemixed input text without being explicitly trained on codemixed examples. We showcase the effectiveness of our method by reporting on part-of-speech tagging and morphological prediction on 70 treebanks and 48 languages.