It seems to me that the ethics here really relate to the foundation upon which you will build. Conducting additional research and writing additional papers is not problematic in itself, but if you must use materials, data, and ideas from informal conversations then you have ethical and/or practical considerations. If you have never met the other researchers and have just read some papers, found it interesting, and then they stopped publishing, I don't see how you would have any obligation to those researchers. This is academic life- we build on the work of others. In fact, you might say we have an ethical obligation to advance the field.
Coming from this perspective, what is acceptable with respect to the foundational elements would be governed by the normal rules. So here are some examples:
- Data: If you need actual (generally unpublished) data from the other researcher(s) then you will have to ask for the data and have their permission to publish that data. You should acknowledge the source of that data in any related publications.
- Materials: If you need code or study materials, you will have to ask for it as well and you will be bound by any relevant licensing restrictions. Again, this should be acknowledged.
- Ideas: If you are building on informal conversations with previous researchers, then you should seek their permission and acknowledge that as well. This is about not losing friends, unless you are under NDA, in which case it may be a legal matter as well.
There may also be cases in which your reputation and friendships may be at stake, for example if you publish research that competes with that of your old advisor. This should not generally be a big concern with projects that are abandoned. If in doubt, you should probably communicate your intentions and seek their blessings.
In many cases, particularly if others have given advice, it may be a good idea to ask if they want to be involved as a coauthor. A researcher who has had to stop doing their research, perhaps due to lost funding or changes in their employment, may be excited at the opportunity to help advance the field while improving their own CV, by just providing some past data and helping with some writing.
The most important thing is to be sure to always give credit where it is due.