When a copyright or trademark license expire, do they automatically turn into public domain?

Since I am nor 100% sure, what you are asking (out of two possibilities) I am going to deal with both possibilities.

An author or a company hold a copyright or a trademark. That copyright/trademark is for a certain number of years unless that protection is somehow extended (and in the past decade or so the period of copyright protection has been extended by Congressional action). Once that period of protection ends, the work goes into the public domain (like many of the pre-1900 works of literature).

While an author holds a copyright, he can grant a temporary license to use that copyright to another company (e.g. grant a movie studio the right to convert a book into a movie within the next ten years). When that license expires, the original copyright is still intact. It's just the permission for someone else to use the work which has expired.

Yes. That is the very definition of an expired copyright. That is also why generic drugs can be sold by different companies after the expiration of the drug trademark.