Generative models have been successfully applied to image style transfer and domain translation. However, there is still a wide gap in the quality of results when learning such tasks on musical audio. Furthermore, most translation models only enable one-to-one or one-to-many transfer by relying on separate encoders or decoders and complex, computationally-heavy models. In this paper, we introduce the Modulated Variational auto-Encoders (MoVE) to perform musical timbre transfer. We define timbre transfer as applying parts of the auditory properties of a musical instrument onto another. First, we show that we can achieve this task by conditioning existing domain translation techniques with Feature-wise Linear Modulation (FiLM). Then, we alleviate the need for additional adversarial networks by replacing the usual translation criterion by a Maximum Mean Discrepancy (MMD) objective. This allows a faster and more stable training along with a controllable latent space encoder. By further conditioning our system on several different instruments, we can generalize to many-to-many transfer within a single variational architecture able to perform multi-domain transfers. Our models map inputs to 3-dimensional representations, successfully translating timbre from one instrument to another and supporting sound synthesis from a reduced set of control parameters. We evaluate our method in reconstruction and generation tasks while analyzing the auditory descriptor distributions across transferred domains. We show that this architecture allows for generative controls in multi-domain transfer, yet remaining light, fast to train and effective on small datasets.