This paper investigates the application of voice activity projection (VAP), a predictive turn-taking model for spoken dialogue, on multilingual data, encompassing English, Mandarin, and Japanese. The VAP model continuously predicts the upcoming voice activities of participants in dyadic dialogue, leveraging a cross-attention Transformer to capture the dynamic interplay between participants. The results show that a monolingual VAP model trained on one language does not make good predictions when applied to other languages. However, a multilingual model, trained on all three languages, demonstrates predictive performance on par with monolingual models across all languages. Further analyses show that the multilingual model has learned to discern the language of the input signal. We also analyze the sensitivity to pitch, a prosodic cue that is thought to be important for turn-taking. Finally, we compare two different audio encoders, contrastive predictive coding (CPC) pre-trained on English, with a recent model based on multilingual wav2vec 2.0 (MMS).