In this work, we induce character-level noise in various forms when fine-tuning BERT to enable zero-shot cross-lingual transfer to unseen dialects and languages. We fine-tune BERT on three sentence-level classification tasks and evaluate our approach on an assortment of unseen dialects and languages. We find that character-level noise can be an extremely effective agent of cross-lingual transfer under certain conditions, while it is not as helpful in others. Specifically, we explore these differences in terms of the nature of the task and the relationships between source and target languages, finding that introduction of character-level noise during fine-tuning is particularly helpful when a task draws on surface level cues and the source-target cross-lingual pair has a relatively high lexical overlap with shorter (i.e., less meaningful) unseen tokens on average.