This is my guess as to what you want...

I worry about data from the four forked processes arriving interleaved on the pipe, maybe even in the middle of each other's lines :(

This solution caches each fork's response until complete and then forwards it to program_A in one piece. I wasn't sure on how to do that using ForkManager.

For testing's sake I combined program_A with the forking program.

#!/usr/bin/perl

# http://perlmonks.org/?node_id=1172353

use strict;
use warnings;
use IO::Select;
$| = 1;

my @data_files = qw( one two three four five six seven );
my $maxchildren = 4;
my %data_for_handles;
my $fh_A;

if( open $fh_A, '|-' )
  { # parent
  warn "pipe opened\n";
  }
else
  { # child
  print "program_A started\n";
  print while <STDIN>;
  print "program_A ended\n";
  exit;
  }

my $sel = IO::Select->new;

while( @data_files or $sel->count )
  {
  while( @data_files and $sel->count < $maxchildren )
    {
    my $file = shift @data_files;
    if( open my $fh, '-|' )
      { # parent
      $sel->add($fh);
      }
    else
      { # child
      $| = 1;
      warn "child $file started\n";
      print "$file\n";
      sleep 1;
      print "$file\n";
      sleep 1;
      print "$file\n";
      exit;
      }
    }
  if( $sel->count > 0 )
    {
    for my $fh ($sel->can_read)
      {
      if(0 < sysread $fh, my $buffer, 16 * 1024 )
        {
        $data_for_handles{$fh} .= $buffer;
        }
      else
        {
        print {$fh_A} delete $data_for_handles{$fh};
        $sel->remove($fh);
        }
      }
    }
  }
close $fh_A;
[download]