I've found three problems so far:
1) Pressing the "insert blind via" key when using the push-and-shove
router does not actually add a blind via.
2) In both "avoid" and "shove" mode, the router treats blind vias as
through-board vias and will move or avoid them even if there's no real
collision (for example, routing a layer 4 trace under a blind layer 1-2
via doesn't work). I've been mitigating this by drawing the portion of
the track that goes under a blind via by hand, but all of the mode
switching is awkward.
3) When a blind via is moved by the router in "shove" mode, it is
converted to a through-board via.