What happens to the mold spores we inhale?
Mold spores are capable of landing and becoming dormant. Unless conditions are severe enough, mold spores will not just simply die off; they will essentially wait for conditions to improve and once those conditions do improve, they will take root and grow. When it comes to the human body, it all depends on the type of mold and the overall health of the host. The body is designed with natural defense mechanisms - mucous, enzymes, beneficial bacteria, filaments, etc. - against airborne particles and microorganisms. Basically, you'll never know if a sneeze or cough - other forms of defense - saved you from a miserable existence as a mold host.

As for whether or not mold spores are expelled from the body, that is not so easily answered. Mold spores can sometimes teem in the millions, especially around a dense infestation (such as grain silos or a house that has structural mold rot). Research has concluded that farmers (who make up 30% of victims who suffer respiratory illness in the U.S. alone), for example, inhale up to 750,000 mold spores per minute. The odds aren't exactly in their favor when it comes to expelling mold spores simply by exhaling.

(I write for MoldBlogger, by the way. If you have anymore questions or are curious about other mold-related scenarios, feel free to check it out.)

Edit: 

Here are some of the sources where a lot of this information can be read about.

National Institute for Occupational Safety and Health. Agricultural Safety. Centers for Disease Control and Prevention website. https://www.cdc.gov/niosh/topics/aginjury/. Published June 19, 2014. Updated December 15, 2014. Accessed February 1, 2017.

Glen H. Hetzel, et al. Farmer’s Lung: Causes and Symptoms  of Mold and Dust Induced  Respiratory Illness. National Agriculture Safety Database website. http://nasdonline.org/1853/d001796/farmer-039-s-lung-causes-and-symptoms-of.html. Published 2005. Accessed February 1, 2017.

Dennis J. Murphy. Agricultural Safety and Health: Farm Respiratory Hazards. PennState Extension website. http://extension.psu.edu/business/ag-safety/health/e26. Published 2017. Accessed February 1, 2017

Also, please feel free to check out a simple post that summarizes this type  of information: Occupational Respiratory Diseases: The Farmer, His Lungs, and Mold