I am aware this doesn't look like elementary-specific but more like a Firefox specific thing, but in fact I only see this with Firefox in elementary OS Loki. It didn't happened in Freya, it doesn't happen with the same Firefox version (49.00) on another ubuntu-16.04-based system I have, and it doesn't happen in other browsers Firefox-forks like Palemoon or Icecat on Loki, but it happens in other versions of Firefox that I tested on Loki, namely thhe nightly build.
Firefox is not able to remember the last download location.
Firefox has two options for downloading: a stable automatic location or the option to ask for the download location each time.
I want it to ask me where to download (instead of automatically downloading in the same location) but to offer me the last used location. (e.g. if last time I have downloaded in Pictures, next time I want to see that as first location offered).
This is the expected behavior, was always the case and it is in other systems as I mentioned.
I want to download some images one after the other, from different websites, one from each website. (I search Google images for "maps of Greece" and I get multiple images on various websites.) When I download them by 'save image as', the download location proposed is always my home directory,
~./ and I have to go each time deep into my directory structure where I want them saved, while normally the second time (at least when the image is the same format) the last selected location should be proposed.
Creating a new empty profile: the same happens.
Testing with the nightly version: the same happens.
Even downloading multiple times the same image and choosing a different location, the same location is opened by default again and again: the home directory.
I can change this location and download, but next time I am shown this again, instead of the expected last used location.
I have tested older versions of Firefox in Loki and the problem is absent. So, there is a problem with Firefox 49.0, but I have it in Loki only and not in Linux Mint 18 (16.04-based too) with this version of Firefox.
Firefox version 45.0 and older is NOT affected
Firefox nightly 52.0 is also affected
Firefox 46.0 is already affected - just like 47, 48, 49, 50 and the present nightly 53.0a1.
This only affects Pantheon Files with Firefox version 46 and newer, while other file managers are not affected (as indicated in this answer).
I have made a bug report here.