It really depends on which is the application that you are trying to migrate.

Wordpress i.e. has several plugins that will offer an easy transition between locally hosted content and content delivered via a CDN, taking care to update necessary links in the process.

When using custom apps, there will be certainly some level of pain involved and mainly when moving TBs of data to the CDN.

We used a mixed approach with a customer recently where we first moved from local content, to content on AWS S3 that was mapped locally via an s3fs (so no change that was needed at app level). Having content already on S3, made it easy to create a Cloudfront distribution and then it was just a matter of changing the URLs of elements delivered via CDN from the application. It worked great and was relatively easy to implement.

Happy to follow up on call.