  We introduce a notion of cross-flips: local moves that transform a balanced
(i.e., properly $(d+1)$-colored) triangulation of a combinatorial $d$-manifold
into another balanced triangulation. These moves form a natural analog of
bistellar flips (also known as Pachner moves). Specifically, we establish the
following theorem: any two balanced triangulations of a closed combinatorial
$d$-manifold can be connected by a sequence of cross-flips. Along the way we
prove that for every $m \geq d+2$ and any closed combinatorial $d$-manifold
$M$, two $m$-colored triangulations of $M$ can be connected by a sequence of
bistellar flips that preserve the vertex colorings.
