Users can "be advised when that author's existing works are cited by new articles. Simply click an author's name from search results, and click Set up e-mail alerts from the author's profile:"
Web of Science permits article citation alerts, i.e. the service will alert you when an article cites a particular article. Click on "Searches and Alerts."
Scopus permits author and article citation alerts. For article citation alerts, from the document view, click on citation alert.
Google Scholar
Authors can use the databases in the box to the left to track citations to their scholarship. Google Scholar though permits you to set up alerts for your scholarship. Learn more here.