This article addresses the adaptation of an acoustic-articulatory inversion model of a reference speaker to the voice of another source speaker, using a limited amount of audio-only data. In this study, the articulatory-acoustic relationship of the reference speaker is modeled by a Gaussian mixture model and inference of articulatory data from acoustic data is made by the associated Gaussian mixture regression (GMR). To address speaker adaptation, we previously proposed a general framework called Cascaded-GMR (C-GMR) which decomposes the adaptation process into two consecutive steps: spectral conversion between source and reference speaker and acoustic-articulatory inversion of converted spectral trajectories. In particular, we proposed the Integrated C-GMR technique (IC-GMR) in which both steps are tied together in the same probabilistic model. In this article, we extend the C-GMR framework with another model called Joint-GMR (J-GMR). Contrary to the IC-GMR, this model aims at exploiting all potential acoustic-articulatory relationships, including those between the source speaker's acoustics and the reference speaker's articulation. We present the full derivation of the exact Expectation-Maximization (EM) training algorithm for the J-GMR. It exploits the missing data methodology of machine learning to deal with limited adaptation data. We provide an extensive evaluation of the J-GMR on both synthetic acoustic-articulatory data and on the multi-speaker MOCHA EMA database. We compare the J-GMR performance to other models of the C-GMR framework, notably the IC-GMR, and discuss their respective merits.