In EndNote, I know there's an option to automatically download PDF files for the references you have in your library. Is there any way to do the same thing in some kind of non-commercial software, like for example Mendeley or BibDesk?
The answer here uses proprietary/paid software. I propose a FOSS solution that works reliably for this task.
The cross-platform software
JabRef has integrated fulltext fetchers to download the fulltext for any entry in the library. Import your bib file into Jabref and select a number of entries. Then, from the
Quality pull-down menu, select
Lookup Fulltext documents. However, you still have to manually confirm the download in a pop-up. But this is still just a bunch of clicking 'OK's at the same spatial location (for each pop-up that appears sequentially) with the mouse. Jabref does the heavy-lifting in the back to correctly download the PDF and link it appropriately by matching it to each relevant citation entry.
The latest master build of Jabref is recommended, since there were some recent fixes to the IEEE Fetcher. https://builds.jabref.org/master/ (if IEEE matters to the OP)
As suggested by @embert in the comments to my original question, this is possible with the help of Citavi. After you've imported your references, go to "References" -> "Check availability and find full text" (the process is explained here).
From initial testing, this feature only seems to work when you have the DOI number for the articles. However, when you do, Citavi finds and downloads the PDFs.