I am assuming the CRM has some sort of input/integration points such as a REST API or URL parameters.
Finesse's workflow has the option of triggering when a call is made/receive/answered/etc. There are two different types of actions: 1) Screen pop, and 2) REST API. So, if your CRM has a REST API as an integration point to pass the information, you can invoke that REST API with the data that you want, such as DNIS or some call variable. The same goes for URL parameters.
Yes, we are implementing Finesse with Oracle Service Cloud CRM - both which have REST API features. I am working on a proof of concept that we are able to integrate the two and that they can communicate by passing call data (ie. talk time, hold time) and call variables. Might you have code snippets that you are able to share or any lessons learned/knowledge articles to help us as we are new to Finesse.