Files of lower priority are unprocessed, so the finding can be cut short when files are found in a higher folder.e.g.

my @folders = glob 'input_directory*'; # get/sort all qualifying folde
+rs
my @requests;
for (my ($found, %filedate); $found = pop @folders or exit();) {
    @requests = sort {
      { ( $filedate{ $a } ||= -M $a )
      <=> ( $filedate{ $b } ||= -M $b )
      } glob "$found/" . 'Flat_file.*.txt'
        and last; # reiterate iff none found
}
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If the loop completes having found @requests, they will have been selected from the highest non-empty qualifying directory and be sorted by creation/modification date. (updated)