Timbre transfer aims to modify the timbral identity of a musical recording while preserving the original melody and rhythm. While single-instrument timbre transfer has made substantial progress, existing approaches to multi-instrument settings rely on separate-then-transfer pipelines that propagate source separation artifacts and produce incoherent synthesized timbres across stems. This paper proposes MixtureTT, to the best of our knowledge the first system for flexible per-stem timbre transfer directly from a polyphonic mixture. Given a mixture and a separate timbre reference for each target voice, MixtureTT jointly transfers all stems to the specified instruments through a shared diffusion process. Modeling the dependencies across the per-stem content and cross-stem harmonic, the proposed joint stem diffusion transformer eliminates cascaded separation error, reduces inference cost by a factor equal to the number of stems, and yields more coherent multi-stem outputs. Despite operating under a strictly harder input condition, evaluations on the SATB choral dataset show that MixtureTT outperforms single-instrument baselines on both objective and subjective metrics demonstrating the necessity of dedicated multi-instrument timbre transfer over the naive separate-then-transfer pipelines. As a result, this work confirms that the cross-stem modeling is essential for mixture-level timbre transfer as the proposed joint setting consistently exceeds an equivalent single-stem ablation.