  Change-point models are widely used by statisticians to model drastic changes
in the pattern of observed data. Least squares/maximum likelihood based
estimation of change-points leads to curious asymptotic phenomena. When the
change-point model is correctly specified, such estimates generally converge at
a fast rate ($n$) and are asymptotically described by minimizers of jump
process. Under complete mis-specification by a smooth curve, i.e. when a
change-point model is fitted to data described by a smooth curve, the rate of
convergence slows down to $n^{1/3}$ and the limit distribution changes to that
of the minimizer of a continuous Gaussian process. In this paper we provide a
bridge between these two extreme scenarios by studying the limit behavior of
change-point estimates under varying degrees of model mis-specification by
smooth curves, which can be viewed as local alternatives. We find that the
limiting regime depends on how quickly the alternatives approach a change-point
model. We unravel a family of `intermediate' limits that can transition, at
least qualitatively, to the limits in the two extreme scenarios.
