Is it appropriate to e-mail a researcher asking about progress on a follow-up paper? Recently, I saw an online presentation by a researcher and professor regarding a series of recently published papers. At the end, she mentioned she was working on a follow-up paper, regarding such and such... I have checked for it, and it has still not been published.

Perhaps it was a dead-end, and abandoned. Would it be appropriate to ask the professor about the progess made, and whether it has been abandoned, or when it will be published?
Certainly this is appropriate, especially if you indicate who you are and why you are interested. Researchers are interested in spreading their results far and wide to get citations.

The worst that can happen is that the research is not yet quite publication-ready, and the author prefers not to share it yet, to avoid academic claim-jumping. If so, you may get exactly this explanation, or no answer at all.

The best, conversely, would be that this could actually lead to a fruitful discussion and/or even collaboration.