Does copyright restrict how I can adapt papers when compiling them into a thesis?

I disagree with part of the other answer, in that the copyright holder of a work is not always the creators/authors. For some journals when you publish with them and sign the licensing agreement, you transfer the copyright of your work to the publisher. Then it is up to the publisher to decide how you can reuse your paper. I would advice reviewing the copyright/licensing agreement you signed with your publisher for limitations for reuse, and if still in doubt, contact the publisher.

As stated in the accepted answer, you should check with the publisher. However, this issue is often clarified specifically online. For example, AIP Publishing states on their website:

Q: May I include my AIP Publishing article in my thesis or dissertation?

AIP Publishing permits authors to include their published articles in a thesis or dissertation. It is understood that the thesis or dissertation may be published in print and/or electronic form and offered for sale on demand, as well as included in a university’s repository. Formal permission from AIP Publishing is not needed. If the university requires written permission, however, we are happy to supply it.

Checking with publisher was mentioned, but not how. Factually all large publishers have this "Request permissions" link directly on the page with the paper. Basically, you click on the DOI of your paper and search for this link. Then you are taken to another website, typically copyright.com. There you have to make some choices, such as "yes, I am the author", "no, it's for the thesis", "yes, all the paper", "no, I won't translate". Finally, a price tag pops up. It is typically 0 Euro/US$/whatever, but I've also seen larger values. You agree (or pay, if you are unlucky), then you receive an email that you officially may use this paper for the purposes you stated. Rinse, repeat.

Small publishers typically state the availability of the papers somewhere close to their front page.

If you have not signed the copyright transfer form yet (i.e., the paper is not finally accepted), you are good. If the paper is Open Access (i.e., licensed under CC-BY or similar), you are good. If you have not transferred the copyright, but licensed the content (say, to ACM), you should be good, but better ask via the above copyright.com process.