use strict;
use warnings;

use XML::LibXML::Reader qw( :types );

sub new {
   my $class = shift;
   return bless({
      reader     => XML::LibXML::Reader->new(@_),
      elem_depth => 0,
      buf        => '',
   }, $class);
}

sub get_next {
   my ($self) = @_;
   my $reader = $self->{reader};

   for (;;) {
      return () if $reader->read() != 1;

      if ($reader->nodeType() == XML_READER_TYPE_TEXT) {
         if ($self->{elem_depth} && $reader->depth() == $self->{elem_d
+epth} + 1) {
            $self->{buf} .= $reader->value();
         }
      }

      elsif ($reader->nodeType() == XML_READER_TYPE_ELEMENT) {
         if ($reader->name() eq 'elem') {
            $self->{elem_depth} = $reader->depth();
         }
      }

      elsif ($reader->nodeType() == XML_READER_TYPE_END_ELEMENT) {
         if ($reader->name() eq 'elem') {
            return substr($self->{buf}, 0, length($self->{buf}), '');
         }
      }
   }
}

{
   my $reader1 = __PACKAGE__->new(location => "file1.xml");
   my $reader2 = __PACKAGE__->new(location => "file2.xml");

   for (;;) {
      my $text1 = $reader1->get_next();
      my $text2 = $reader2->get_next();

      last if !defined($text1) && !defined($text2);
      die if !defined($text1);
      die if !defined($text2);

      process_data($text1, $text2);
   }
}
[download]

Assumes all elem elements are "interesting" ones, not just the ones found under the root. Easy to change, though.

Output left to the user. May I suggest XML::Writer since it keeps next to nothing in memory.