This method appears to follow these rules:

 - if $localName names a sibling node, the cursor is moved to that node;
 - if $localName names an ancestor node, the cursor is moved to the end of that node;
 - if $localName names a node that is a sibling of any of the current node's ancestors, the cursor is moved to that node;
 - otherwise the cursor is moved outside the document.

Note especially that this method never moves the cursor to child nodes.

For example, given this XML document

<?xml version="1.0" encoding="UTF-8"?>
<root id="root" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">
  <aaa id="1">
    <bbb id="1.1"></bbb>
    <bbb id="1.2"></bbb>
  </aaa>
  <ccc id="2" />
  <aaa id="3">
    <ddd id="3.1">
      <aaa id="3.1.1"></aaa>
    </ddd>
    <aaa id="3.2"></aaa>
  </aaa>
  <aaa id="4">
    <eee id="4.1"></eee>
  </aaa>
</root>

going from <root id="root"> to "bbb" places the cursor outside the document;
going from <aaa id="1"> to "bbb" places the cursor outside the document;
going from <aaa id="1"> to "aaa" places the cursor on <aaa id="3">;
going from <bbb id="1.1"> to "bbb" places the cursor on <bbb id="1.2">;
going from <bbb id="1.2"> to "bbb" places the cursor outside the document;
going from <bbb id="1.1"> to "ddd" places the cursor outside the document;
going from <bbb id="1.1"> to "aaa" places the cursor on </aaa>;
going from <bbb id="1.1"> to "ccc" places the cursor on <ccc id="2">;
going from <bbb id="1.1"> to "nonsuch" places the cursor outside the document;
going from <bbb id="1.1"> to "root" places the cursor on </root>;
going from <ddd id="3.1"> to "aaa" places the cursor on <aaa id="3.2">;
going from <ddd id="3.1"> to "eee" places the cursor outside the document;

Try it yourself:

<?php

$document = <<<XML
<?xml version="1.0" encoding="UTF-8"?>
<root id="root" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">
  <aaa id="1">
    <bbb id="1.1"></bbb>
    <bbb id="1.2"></bbb>
  </aaa>
  <ccc id="2" />
  <aaa id="3">
    <ddd id="3.1">
      <aaa id="3.1.1"></aaa>
    </ddd>
    <aaa id="3.2"></aaa>
  </aaa>
  <aaa id="4">
    <eee id="4.1"></eee>
  </aaa>
</root>
XML;

$filename = "/tmp/xmlreader.php.xml";
file_put_contents($filename, $document);

echo "given this XML document\n\n$document\n\n";

showNext("root", "bbb");
showNext("1", "bbb");
showNext("1", "aaa");
showNext("1.1", "bbb");
showNext("1.2", "bbb");
showNext("1.1", "ddd");
showNext("1.1", "aaa");
showNext("1.1", "ccc");
showNext("1.1", "nonsuch");
showNext("1.1", "root");
showNext("3.1", "aaa");
showNext("3.1", "eee");

function showNext($from, $to) {
  global $filename;

  $xml = new \XmlReader();
  $xml->open("file://$filename");
  while ($xml->read()) {
    if ($xml->nodeType === \XmlReader::ELEMENT) {
      if ($xml->getAttribute("id") == $from) {
        echo "going from <$xml->name id=\"$from\">";
        break;
      }
    }
  }

  $xml->next($to);
  $destination = "";
  if($xml->nodeType === \XmlReader::NONE) {
    if(!$xml->read()) {
      $destination = "outside the document";
    }
  }

  if(!$destination) {
    if ($xml->nodeType === \XmlReader::END_ELEMENT) {
      $destination = "on </$xml->name>";
    } else if ($xml->nodeType === \XmlReader::ELEMENT) {
      $destination = "on <$xml->name id=\"" . $xml->getAttribute("id") . "\">";
    }
  }

  echo " to \"$to\" places the cursor $destination;\n";

  $xml->close();
}
?>

next() without parameters will bring you to the next sibling node on the same depth as the depth where the current cursor is. 
Examples:
1. If you are on an opening ELEMENT node next() will bring you to the new line node next to the closing tag of that element (i.e. if you have each tag in separate line) or it will bring you to the opening tag of next node on the same depth.
2. If you are on a TEXT node next() will bring you to the opening tag if there is one next to it.

If there are no more nodes on the given depth next() will bring you to the closing tag (END_ELEMENT) of the wrapping parent.

To skip over deletion nodes in the xml extracted from a word document, do something like this:

if ($paragraph->nodeType == XMLREADER::ELEMENT && $paragraph->name === 'w:del'){$paragraph->next();}