We provide examples of families of (log) smooth canonically polarized varieties, including smooth weighted pointed curves and smooth hypersurfaces in with large degree such that the Chow semistable limits under distinct pluricanonical embeddings do not stabilize.
"Nonexistence of asymptotic GIT compactification." Duke Math. J. 163 (12) 2217 - 2241, 15 September 2014. https://doi.org/10.1215/00127094-2785806