It depends on whether youre dayside or O/N BR. I can only speak for dayside (at an O/N truck unload store). At the open, you help out O/N with whatever backstock maybe leftover from the truck, then priority moves to pulling Instock's research batches, pulling plano's POG's and salesfloor POG's (tho the guy who dropped it sometimes pulls it; we pull it as courtesy), and pulling PTM's. At 11am, the hourly CAF batches drop on the top of each hour, until 5 or 7, varying on location. This is now the prime priority to finish before the next hour drops (which during Q4 means youre getting an awesome workout 😉 ). In between CAF's, you will backstock, and run misc. tasks, usually GS requests and pulling a POG or EXF. Later in the day, you will pull Price Changes (6pm at my store, dunno if this is true everywhere) for the pricing team to ticket the next morn. Closing BR involves heavy backstocking, killing and condensing pallet spaces, and setting the line for O/N, if you know how to. Sorry if it's a lot to take in, but that's the gist of it.
Also, about pushing, it is NOT in your core roles as BR to push, however, during certain parts of the year and in lower-volume stores, you're bound to do it sometime just to escape boredom.