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The integration of multimodal data presents a challenge in cases when the study of a given phenomena by different instruments or conditions generates distinct but related domains. Many existing data integration methods assume a known one-to-one correspondence between domains of the entire dataset, which may be unrealistic. Furthermore, existing manifold alignment methods are not suited for cases where the data contains domain-specific regions, i.e., there is not a counterpart for a certain portion of the data in the other domain. We propose Diffusion Transport Alignment (DTA), a semi-supervised manifold alignment method that exploits prior correspondence knowledge between only a few points to align the domains. By building a diffusion process, DTA finds a transportation plan between data measured from two heterogeneous domains with different feature spaces, which by assumption, share a similar geometrical structure coming from the same underlying data generating process. DTA can also compute a partial alignment in a data-driven fashion, resulting in accurate alignments when some data are measured in only one domain. We empirically demonstrate that DTA outperforms other methods in aligning multimodal data in this semisupervised setting. We also empirically show that the alignment obtained by DTA can improve the performance of machine learning tasks, such as domain adaptation, inter-domain feature mapping, and exploratory data analysis, while outperforming competing methods.