I changed twice (!).
Started at one flying school then FI lost his medical and there were no replacements. So I changed schools, but then after all theory exams and QXC for entirely work pressure related reasons took a break for a short while.
A few months later original FI was back in the air, and after talking to both I finalised my training and took skills test with first school.
Both CFI's certified the relevant parts of the logbooks pertinent to their time of training, and certified copies of the training records were exchanged each time so effectively the 'new' FI each time knew what had been covered by the previous. The CAA accepted this. Neither training organisation saw it as unusual.
So far as experience goes, one was a grass strip, the other a busy commercial airport so I feel I had the best of both worlds.
Undoubtedly, however, the change and the break in training were a major reason why my qualifying hours were higher than I'd originally hoped for which needs to be borne in mind when deciding to change.
All I would add is the importance of maintaining relationships. If I'd fallen out at either stage I wonder how the request for certifying the logbook would have gone down.
This was in 2013, the regulations may have changed since then of course.
Good luck