perl -0777 -pe's/^.*\n//' any number of files
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This won't work correctly on files with a single line and no terminating newline, though. If that is a concern, the following is necessary:

perl -0777 -pe's/^.*(?:\n|\z)//' any number of files
[download]

I really wish there was a metacharacter for "either newline or end of string" as the counterpart to $ — this isn't the first time it would have been handy.