Here’s the workflow that we found works for title transitions in Avid - the third paragraph in the text below is most pertinent to your question from above (I recommend saving this out as a text file for future reference.)
The most important thing is that both clips that are transitioning must have either a BCC or Sapphire filter on them and be set with background None so they preserve the alpha. Then apply the Continuum or Sapphire transition. Then use BCC Composite on the track above to do the real composite over the background.
Transitions between BCC Titles: Create two separate overlapping title clips on V1 with any BCC filter such as Title Studio. Make sure the Background layer is set to None in both the titles. Then apply any BCC or Sapphire transition to the overlap between the two titles to create a transition between two titles which appears to be composited over black but which has a live alpha channel. Then add the desired final background clip to V2 (above the transition) and apply BCC Composite to this clip. Change the composite mode to Normal instead of default and you should see the transition between titles composited over the unaffected background.
Transitions between native title-mattes: Take the same approach as for transitions between BCC Titles (above) but requires a few additional steps. Create native avid title mattes on the two V1 clips. Apply BCC Brightness and Contrast effects to each native title matte, select Apply to Title, and set Background to None. This will preserve the alpha channel so that when the BCC or Sapphire transition is applied, the final Composite filter can generate a true composite between titles with intact alpha channels.