Almost. Your code is not qualifying the state before changing it.

Maybe like this:

my @fn = (\&doOtherProcessing, \&doStage1, \&doStage2, \&doStage3);
my $fn = 0;
while (<DATA>) {
    if ($fn == 0 and /for/)   {$fn = 1}
    if ($fn == 1 and /their/) {$fn = 2}
    if ($fn == 2 and /fruit/) {$fn = 3}
    if ($fn == 3 and /lazy/)  {$fn = 0}
    $fn[$fn]->($_);
}
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