BusinessProcess type should be used in the package.xml file to retrieve or deploy Salesforce Case Support Processes using Apache ANT Tool.

Sample package.xml:

<?xml version="1.0" encoding="UTF-8"?>
<Package xmlns="http://soap.sforce.com/2006/04/metadata">
    <types>
        <members>Case.Standard</members>
        <name>BusinessProcess</name>
    </types>
    <types>
        <members>Case.Internal</members>
        <members>Case.External</members>
        <name>RecordType</name>
    </types>
    <types>
        <members>Testing</members>
        <name>Profile</name>
    </types>
    <version>54.0</version>
</Package>