I fly a classic 747, which is even older technology, and we don't run the alerter down to the field elevation for an RNAV or RNAV/VNAV approach.
We do follow the altitudes down for each step fix, even though we're doing it as a constant descent, with the alerter. We fly it either with the manual pitch trim wheel (my preference) or off the vertical speed wheel, and follow a vertical path deviation indicator ("chinese glide slope).
We train for both constant descent approaches and for dive and drive, and the preference is to make all procedures that aren't ILS, that can be called up in a database, into constant descent. Accordingly, we fly the VOR or NDB (which we do train for and demonstrate) the same as a basic RNAV.
Our alerter and radar altimeters are set to MDA plus 50'. Any other time, precision approach or a step-down "dive and drive," the alerter is set to the missed approach altitude after we're done stepping down, or in the case of an ILS once glideslope intercept/capture has occurred.