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You can easily set up an ongoing search and receive updates when new articles cite an article of interest.
In Google Scholar, under the article you're interested in, click on the "Cited by" link.
Then when you're looking at the list of documents that have cited the article, on the left you'll see a link to "Create alert." Clicking on that will track citing articles and send you an email every time there's a new one.
In Web of Science, click on the title of the article to see the full details. The "Citation Network" will display on the right. Click on "Create citation alert" to be notified by email when a new publication in Web of Science cites the one you're viewing. You'll need to create a free account in Web of Science to use this feature.
In Semantic Scholar, you can set up a free account. Then when you find an article you're interested in tracking, you can easily set up a citation alert. Click on "Alert" underneath the reference. Then you'll receive an email every time a new article cites your article of interest.