How do you handle updates or modifications to Lottie animations after they've been deployed on a website or app, ensuring the new versions are compatible with the existing code?

There can be Two Scenarios for this:

1) Most Recommended to add the Lottie JSON File Directly into your Media Library of your website CMS using Lottie Plugin ( if supported for that specific CMS like Webflow ). 

2) If your web is on local server or not have direct Lottiefiles Plugin access then second case to use the Lottie Shortcode to embed into the backend files to make the animation live on web. ( In that case, I would highly recommend to use the Player JS file alongside with the Lottie JSON file without relying on the package ). But still if you go to add package or dependency CDN for Lottie then it would work fine for future also, there will be no issue with that.