  We derive a continuum mean-curvature flow as a certain hydrodynamic scaling
limit of Glauber-Kawasaki dynamics with speed change. The Kawasaki part
describes the movement of particles through particle interactions. It is
speeded up in a diffusive space-time scaling. The Glauber part governs the
creation and annihilation of particles. The Glauber part is set to favor two
levels of particle density. It is also speeded up in time, but at a lesser rate
than the Kawasaki part. Under this scaling, a mean-curvature interface flow
emerges, with a homogenized `surface tension-mobility' parameter reflecting
microscopic rates. The interface separates the two levels of particle density.
  Similar hydrodynamic limits have been derived in two recent papers; one where
the Kawasaki part describes simple nearest neighbor interactions, and one where
the Kawasaki part is replaced by a zero-range process. We extend the main
results of these two papers beyond nearest-neighbor interactions. The main
novelty of our proof is the derivation of a `Boltzmann-Gibbs' principle which
covers a class of local particle interactions.
