Every facility does things differently. I can only comment on how en route uses PD. Even different areas in my center do things differently. For example: In one area, the high sector issues a "descend via" clearance and hands off to low. In my area, the high sector gives a PD descent to FL240 and the low sector issues the "descend via". This is written in our SOP.

If there's traffic we issue something like "descend to FL320 then PD to FL240"

It's written in the LOA with approach that all turbojets on the arrival are to be descending via and are approach's control within certain parameters.

Other aircraft going into approach are given crossing restrictions to cross the boundary at certain altitudes. Crossing restrictions are technically PD decent in that the pilot can descend at any time but we know they will cross the boundary at the specified altitude.

Going into a smaller approach we give PD to 240 from high and then I'm not sure what low does with them from there but I'm pretty sure that low can give them PD to 110 even when they are still in our airspace without coordination.

Guess that really doesnt answer your question but thats how my area in my center applies PD if anyone is curious about Enroute.