I permanently live abroad with my spouse. My passport was due to expire, so I sent it, along with the necessary renewal form, picture, etc, back to the US to be renewed. I'm planning on having my family send it back to me. Since there is no exit stamp from the country I now reside in, will this cause an issue for me? What will happen with my passport and renewal that I already sent in?
When the State Department issues you the new passport, they will also return the old one (with some holes punched through it to invalidate it). In most instances, you can simply show the stamp in your old passport, along with your new passport.
IIRC, though, there are a few places that require stamps to be transferred to the new passport. So you may want to contact the immigration service of whatever country you're in, just to make sure.
EDIT: RedBaron suggests in a comment that you might be concerned that the US won't properly process your renewal because you don't have an exit stamp. If so: Don't worry. The US does not care about exit stamps, certainly not while processing a passport renewal. (In my own passport, which has since been renewed, I had about three missing exit stamps.)