This extracts all the childnodes into an array. I've printed them out with numbers and delimiter so you can see them individually. I suggest you loop through them extracting all or any that you want.

#!perl
use strict;
use XML::LibXML;

my $dom = XML::LibXML->load_xml( IO => *DATA );
my $study_str = 'Purpose and rationale|Study purpose|Study rationale';
for my $search ('//header') {
  my $nodeset = $dom->find($search);
  foreach my $node ($nodeset->get_nodelist){
    if ($node =~ m/$study_str/i){
      my @childnodes = $node->nonBlankChildNodes();
      my $n=1;
      for (@childnodes){
        print '#'.$n++.'# '.$_->toString."##\n\n";
      }
    }
  } 
} 
__DATA__
<root>
<part><sect>
 <header>
1. Purpose and rationale doc 1
<p>purpose 1</p>
  <p>Purpose 2</p>
  <p>purpose 3</p>
  <ul>
   <li>purpose list 1</li>
   <li>list2</li>
  </ul>
  <p>2. Some other heading</p>
  <p>content 1</p>
  <p>content 2</p>
 </header>
 </sect></part>
</root>
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