These sequences date back (at least) to Enrico Ducci (1864-1940).
Results about such sequences have been reformulated and reproved
several times over the last century, so care must be taken so that
history does not keep repeating itself.
My papers on this topic follow. The second paper has a good bibliography
of papers up to 2003.
M. Chamberland and D. Thomas (2004),
The N - Number Ducci Game - Open Problems,
Journal of Difference Equations and Applications,
10 , 339-342
M. Chamberland (2003),
Unbounded Ducci Sequences,
Journal of Difference Equations and Applications
9 , 887-895
O. Andriychenko and M. Chamberland (2000),
Iterated Strings and Cellular Automata,
Mathematical Intelligencer 22(4), 33-36
Other people who work on such problems include: