Two major issues in my experience seem to occur.

The first is that you had a system with a given set of ownership and now lines need to be drawn between groups to grant each sub-team their own piece of the larger pie. This is where Conway's law comes to bite you because your code is likely structured around your existing team and practices. Deciding how to draw that boundary is a challenge (API-based? Separate services?). Do not skip this part, otherwise you'll have an awful mix of old and new and everyone suffers.

The second is how work is structured. With a small team, anyone can edit anything (ownership again). With multiple teams you need to accept that changes will require multiple stages of development and the rate of change can take a hit due to scheduling and prioritisation for each team. In the small team a single sprint (assuming this is your working practice) may have been sufficient, but with multiple teams those changes will need scheduling.