One possible way* a collision with another universe could affect ours is by producing a large-scale anisotropy in the universe as a whole. Researchers recently found a "cold spot" in the cosmic microwave background that has not yet been explained by processes known to physics within the universe (https://arxiv.org/pdf/1704.03814.pdf), meaning that it could be the result of such an event (in the sense of "there's a nonzero, but still very small, probability that this is the correct explanation").
I have not heard of any arguments** linking the accelerated expansion of the universe to collisions with other universes.
Even the existence of the dark flow is still hotly debated; for example, observations from the ESA's Planck satellite are not consistent with dark flow observations made earlier (https://arxiv.org/abs/1303.5090). If dark flow exists, NASA's Goddard Space Flight Center's Alexander Kashlinsky says that we don't have enough information to count it as evidence of another universe, let alone specifically a collision (https://www.seeker.com/mysterious-dark-flow-may-be-tug-of-other-universe-1765035964.html).
*There are probably many other ways, but I've only read a small fraction of the cosmology theory papers out there, and I've understood an even smaller fraction. Nearly every question that starts with "Could" or "Can" in physics has the answer, "Yes, in some theorist's version of reality," and any question starting with "How" can be answered in a myriad of ways depending on the answerer's preference.
**Again, I've only read so many cosmology papers, so the argument may indeed be out there.