There are several cell processes involved in endosymbiotic theory, some of them more directly than others.
Endosymbiotic theory suggests that mitochondria and chloroplasts, rather than being derived from other eukaryotic cell structures, were once separate, independently living prokaryotes. Analogues for chloroplasts are found in modern cyanobacteria, which are capable of photosynthesis, and analogues for mitochondria are found in proteobacteria. Some of the primary evidence for this theory is that the chloroplast and mitochondria contain significant amounts of their own, independently replicating DNA, but not enough to allow them to live independently of the cell environment; instead, these controls are in the nuclear DNA. This suggests that the mitochondrial and chloroplastic DNA was progressively stolen by the nucleus in order to ensure dependency. This process would be the one truly responsible for the "evolution" of mitochondria and chloroplasts as we know them from their prokaryotic ancestors.
The cell process that would have initiated endosymbiosis is endocytosis; the process of taking things inside the cell. The theory suggests that the ancient prokaryotes may have been eaten by the ancient eukaryote, but that the eukaryote failed to kill or digest them.
Another process is simply the metabolism itself; the primary benefit of the mitochondria and chloroplast are that they allow the cell to increase the amount of energy available for metabolic reactions.