Pre-trained multilingual models have enabled deployment of NLP technologies for multiple languages. However, optimally fine-tuning these models under an annotation budget, such that performance on desired target languages is jointly maximized, still remains an open question. In this paper, we introduce DeMuX, a framework that prescribes the exact data-points to label from vast amounts of unlabelled multilingual data, having unknown degrees of overlap with the target set. Unlike most prior works, our end-to-end framework is language-agnostic, accounts for model representations, and supports multilingual target configurations. Our active learning strategies rely upon distance and uncertainty measures to select task-specific neighbors that are most informative to label, given a model. DeMuX outperforms strong baselines in 84% of the test cases, in the zero-shot setting of disjoint source and target language sets (including multilingual target pools), across three models and four tasks. Notably, in low-budget settings (5-100 examples), we observe gains of up to 8-11 F1 points. Our code is released here: https://github.com/simran-khanuja/demux.