PE Analyst -> PE Associate at another firm? Is this possible?

If you were asking this question five years ago it would be somewhat of a different story. Today the reality is that enough well-established firms (even beyond the megafunds like Blackstone, TPG, and KKR that really led on this) have ventured into taking analysts directly out of undergrad that there's a critical mass of professionals who began in one of them and eventually moved to a different firm.

I just made a quick comment a moment ago in another thread that was probably posted by the first user to reply here. Here's the relevant portion:

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The smartest way to do this is to find out what headhunters the firms you're interested in use, then email them around one year into your current role to set up phone and coffee chats.

For banking analysts, they'll do this about two months after getting on the desk in order to be positioned for the absurdly early on-cycle schedule that now seems to go a whopping total of four months after the analyst start date.

In your case, you'll want to go on-cycle a year later, as in 16 months after your start date. This will allow you to recruit both for proximate opportunities (the summer after your recruiting period) and on-schedule (for two summers away).

Personally, I wouldn't recruit as a first-year. There are enough slots at great shops (megafunds included) available for same-year start that you could safely go as a new second-year and not worry about not being able to find a job to start in June. Some of the big funds have even begun intentionally reserving a couple spots in each class for second or third-year analysts who choose to recruit later.