Unfortunately, only google docs and folders can have ownership changed, as per this help page, and if you try to change another file, (e.g, a zip or pdf) you'll get an error like 'you can't yet change the owner of this file, we're working on it'. select 'Make owner' (if different items currently have different owners, it will say 'various', but will still give you the option to assign a new owner).select the new user to who you wish to assign ownership.right-click all selected items, select 'Share'.Select them all (eg, 'ctrl-a' to select all, or shift-click top & bottom items) the resulting file list is all files and folders, recursively beneath that folder.'right-click' on the top-level folder on which you want to recursively change ownership (OR, in the full directory path shown at the top, click the 'down arrow' on the current folder), and select option 'Search within '.A simple method (and slightly different than other suggestions currently posted) to recursively change ownership of all items within a folder:

Dropbox transfer ownership