Last week I've change my password though the website, at work. On my home computer I have installed the Desktop client. I have not changed a thing and it was syncing without problems.
The same happened with my Dropbox App for my Android phone.
If I ever change the password of my account I was expecting the need to update it everywhere I use it. If, by any change, somebody uses a Desktop/Mobile client and I change my password, this person would be able to keep on using it without problems.
From my point of view, unless I'm missing something, this is a security flaw that must be corrected.
Hope to hear from Dropbox team.