It sounds like you'd be fine with processing them one at a time, not all at once. In that case, just wrap your while loop in another loop that goes through the files:

my @files = qw( /path/to/file1
                /path/to/file2
                /path/to/file3
               );

for my $file (@files){
  open my $fd, '<', $file or die $!;
  while(my $line = <$fd>){
      # do stuff with the next line
  }
  close $fd;
}
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