Central Library for Multiple Applications?

I’ve recently started a position at an established company that didn’t have a single interactive design lead. I tasked myself with creating a centralized library of design elements. Then, I learned that the design agency that redesigned our public website last year didn’t hand over design files, and every designer is working in a different application. As this is now ‘established’, and the company is very against micro-managing, I now have a much more complicated task.

So far, one UX designer uses XD primarily, as do I. The pure UI designer uses Sketch. A freelancer who’s been around long enough to sort of be a manager has made himself a Figma library for his specific area of our site. And the UX manager that I’m now partners with uses Axure a lot. As it is, the dev department has components already coded, and the UI guy often mocks up screenshots in PS and edits existing code in the browser and screenshots that. Everyone is doing their own thing.

So, I’m looking for advice from anyone who’s done something like this. I remember importing Sketch into XD renamed everything and made existing prototypes almost useless. I’ve only dabbled in Figma, and I’m not expecting Axure to be awesome for importing graphics, but I haven’t tried in a very long time.

At the worst, I’d like to have a source file for each program, and these might need to be updated manually whenever we change or add assets. Has anyone successfully accomplished something like this?