Does the A320's FMC automatically adjust the rate of descent on a non-precision approach?
Your mentioning of LOC BC is interesting.

The answer is yes except for LOC or LOC BC approaches.

When the FMGS guides the plane down an approach, it is called managed. If manually selected (e.g. vertical speed or FPA), then it's called selected.

An A320 can fly a fully managed (lat/vert) NPA if it is VOR, VOR/DME, RNAV, or NDB (given that the approach is correctly intercepted and GPS primary is available or the navigational accuracy is good). If all engagement conditions are met, the 2nd and 3rd columns of the FMA will merge and read FINAL APP (you can watch this switch here), and will be flown down to the MDA, after which (by 50 feet) the AP disconnects.

Note: the coded approach needs to be validated and approved by the operator.


  


If the NPA is LOC, the official SOP is to fly it in LOC mode for lateral and selected for vertical – while calling out the raw data. That's because in a LOC approach the vertical deviation (that appears on the PFD in place of the glide slope) must be ignored because the MAP (missed approach point) could be before the runway.*

* No other reason is given, which may hint at a database limitation (possibly for safety reasons). Boeing's flavor, IAN (which is newer), allows G/P vertical guidance for LOC and B/CRS approaches.

Reference: A320 FCOM, SOP § 3.03.19.



I checked the A380, and its newer FLS (FMS Landing System) allows vertical guidance for LOC and LOC BC approaches, but with a caveat similar to the A320 (but in a more clear manner):


  However, the FLS function is not available, if the NPA is as follows:
  
  
  The Missed Approach Point is before the runway threshold, and
  The difference between the final approach course and the runway course is more than 50°.
  
  
  When the flight crew selects an NPA, and if the FLS function is not available for this NPA, the NO FLS FOR THIS APPR message appears on the FMS message area. [A380 FCOM § 22-FG-70-80-30]