Captions are managed in the catalog. That's true for other types of data entered in the catalog like keywords. The principle is to choose what and when you want to have captions 'written in the metadata header' section of your photo files (or in sidecar xmp files for raws).
So, you have to choose your files (that can be a huge batch) and 'write metadata to files' (shortcut Ctrl V).
Note that captions added individually should be written automatically to files, not so with batch addition of captions. Captions are entered in different fields of the metadata (eg IPTC and xmp) at the same time. That may explain incompatibility betwween various management tools.