XML::Twig is especially designed to work with big files. What you want to do can be done with the following code:

#!/usr/bin/perl -w
use strict;
use XML::Twig;

my $t= XML::Twig->new( twig_handlers => 
        { node => sub { my( $t, $node)= @_;
                        return unless( $node->has_children( 'child'));
                        my $name= $node->first_child( 'name');
                        my $file= $name->text;
                        $name->delete;
                        print "in file $file\n"; # in real code open f
+ile
                        $node->print;            # in real code print 
+to file
                        print "\n\n";
                        $t->purge;               # get rid of the node
                      }
       })->parse( \*DATA);
                          

__DATA__
<root>
  <name>Parent Name</name>
  <node><name>Node 1</name>
  <node><name>Node 2</name>
    <child id="2263583">
      <start>
        <thing value="20001231945"/>
      </start>
    </child>
  </node></node>
  <node><name>Node 3</name>
  <node><name>Node 4</name>
    <child id="2274693">
      <start>
        <thing value="20001231930"/>
      </start>
    </child>
  </node>
 </node>
</root>
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