The airprox (I hope you reported it) was probably not down to the changing of units - under a RIS you should ideally be getting handed-over unit to unit rather than a "freecall", and both controllers should be looking at your return & any adjacent ones. The problem is, I suspect, likely to be one of a/c type and flight profile, with a bit of radar equipment thrown in. In general -
The smaller & less-metallic an a/c, the harder it is for the primary radar to see.
tangential fade/critical relative speed - primary radar won't see it.
No transponder - SSR won't see it
If synthetic (processed) radar loses a target (eg if it's manoeuvring) it will produce a return for where it thinks the a/c should be, rather than for where it is.
The only real way to get to the answer is to file an airprox & have the incident investigated. Have to say, though, top marks for always using the best service you can get. I get so many people calling me and blithely announcing "squawking standby" - who's that helping?!
ap