There are no issues carrying firearms to or from airports with air carrier service.
The issue is more one of local ordinances; for example if you depart Elkhart, IN with a LTCH and your trusty Glock 23 in a well-worn Crossbreed and land at MDW, you will be in violation of Chicago/Cook County law.
If you instead went to FWA, SBN, IND, DTW or CLE then you wouldn't have any legal issues as those localities don't have a ban on firearms or more specifically, handguns.
Same deal with "assault rifles"; I could easily put a Colt 6920 with a case of Magpul 30s in the Citation and fly from CLE to BGR, but if I stopped in ROC then I would be in violation of the NY SAFE Act.