Counting the the number of publications citing all of a given set of publications I would like to specify a list of publications and find out which articles cited the ones in my list.

Is there such tool?
You could probably use Europe PMC API for that. It would be limited to biomedical publications though. Using this module you should be able to construct your query and find intersection programmatically. Alternatively you can combine the queries on the website using Boolean logic, e.g. here is a search for a paper citing three particular publications.

Disclaimer: I work for Europe PMC