[Boomer]

You might want to check into doing direct rollovers from those accounts into IRAs — like Fredman and retiredguy123 said here.

I prefer IRAs to leaving the money in the 401(k) or 403(b) because IRAs give more direct control.

If you do this, be sure you do it absolutely perfectly so that you do not trigger any tax consequences.

Regarding your original question:
I am wondering if you decide to consolidate if you will be able to keep all of your current funds. Individual stocks would roll over directly. But I do not know if mutual funds titled specifically to the firm could roll over directly to a different firm. For example, could Fidelity Contrafund roll over to Vanguard as is? (I do not know the answer to that but it might be no. Maybe somebody here knows.)