My question was usefully answered in the comments: Otis Chodosh pointed me to the inverse mean-curvature flow as described, e.g., in this paper:

Huisken, Gerhard, and Tom Ilmanen. "The inverse mean curvature flow and the Riemannian Penrose inequality." Journal of Differential Geometry. 59.3 (2001): 353-437.

and Willie Wong explained how it would evolve dumbbells. Rbega pointed me to this paper

Kazhdan, Michael, Jake Solomon, and Mirela Ben‐Chen. "Can Mean‐Curvature Flow be Modified to be Non‐singular?." Computer Graphics Forum. Vol. 31. No. 5. Blackwell Publishing Ltd, 2012.

which contains this image of their modified "conformalized mean-curvature flow" applied to dumbbells: