We provide a new type of proof for the Koebe-Andreev-Thurston (KAT) planar circle packing theorem based on combinatorial edge-flips. In particular, we show that starting from a disk packing with a maximal planar contact graph $G$, one can remove any flippable edge $e^-$ of this graph and then continuously flow the disks in the plane, such that at the end of the flow, one obtains a new disk packing whose contact graph is the graph resulting from flipping the edge $e^-$ in $G$. This flow is parameterized by a single inversive distance.