By a topological (2,2)-flip Dougherty, Faber, and Murphy mean a bistellar move on 
2-manifolds (and not on 3-manifolds). So it looks like the example

>on a 3-simplex, a (2,2) non-topological flip would map one edge of the tetrahedron into the edge it does not intersect.

is supposed to be about trying to apply a bistellar move to the *boundary* of a 3-simplex. It further looks like the geometers want to see the result of such a generalized bistellar flip as the 2-disk, triangulated by the suspension over the 1-simplex. As a topologist, I'm deeply troubled by such vision of a 2-dimensional Pachner move (e.g. what of a "flip" does it retain if so generalized?) I would either think of this as a 3-dimensional move indeed (see more about this below) or else I would consider the following to be the result of this move: two copies of the suspension over a 1-simplex, glued along their boundaries. This is no longer a simplicial complex, but a "pseudo-complex" in the sense of the Hilton-Wylie textbook, and a "singular triangulation" of a more modern tradition. 

Certain generalizations of bistellar moves to singular triangulations have been studied by Matveev and his students; they are precisely dual to Matveev's moves on special spines (concerning the duality, see [this review][1], though there must be better references). Some of these generalized bistellar moves are not supported by homeomorphisms (so maybe geometers would call them "non-topological"). For instance, there is a move that collapses the join of a 1-simplex and S, where the 1-sphere S is the union of two copies of a 1-simplex along their boundaries, onto the suspension over a 1-simplex. This 3-dimensional singular flip can be decomposed into a sequence of two "(2,2) non-topological flips".

  [1]: http://www.ams.org/mathscinet-getitem?mr=2012938