'following the updates' is correct and usable in written English.
You could use it to refer to keeping track of or staying informed about any changes that have been made. For example: "I like to stay up to date with my favorite TV show, so I'm always following the updates on the show's social media page."
Just got into the office but can't start working because am following the updates.
Twittering an operation might seem frivolous, but when Ms. Nelson's teeth began chattering after the procedure, a friend following the updates suggested it could be a potentially hazardous side effect, tardive dyskinesia, that can occur with one of the antinausea drugs Ms. Nelson was taking.
We are planning to update the database regularly, following the updates of UniProt, as well as to extend its content.
The trial was exclusively developed and performed at the facilities of the University of Extremadura, Spain, with the approval of the Committee on Biomedical Ethics of the University and following the updates of the Declaration of Helsinki.
Following the updates, search results began to include signals from your Google+ Circles.
From Michael Johnson on Twitter: "Following the updates from Buenos Aires Airport, it's 27 degrees here, not looking forward to arriving to LHR tomorrow".
Analysts at Arbuthnot cut their target price from 60p to 25p following the update.
Carphone's shares have jumped 5p to 137p following the update.
Analyst Oliver Wynne-Jates at Panmure Gordon said a major re-rating was in store following the update, and has lifted his target price to 680p.
The Guardian has seen an iPhone locked in a reboot cycle which iTunes will not recognise following the update, while Engadget reported one of its journalists had suffered similar issues.
Infrastructure and maintenance company Mouchel was buoyant on Tuesday ahead of its delayed results, but following the update, it has lost virtually all of the gains and is now down 5p at 12.5p.
