In your downloadable example, the problem seems to be that the elements move "under" the currently displayed content, but still prevent interaction in the area over the embedded things.
On the "official" demo page, the problem can be noticed as well, but they cleverly formatted the first tab so it isn't getting overlepped from the image under it.
EDIT: Testing the live website I found that changing the .js .tabs > section selector's visibility: hidden; to display: none; AND changing .tabs > section[aria-expanded="true"]selector's visibility: visible; to display: block; solved the problem. You could try that. It does seem like that the 'bubbling' issue may come into play, so try both this and the other answer together.
