Working on the fact that it is one ticket and routed via Chicago and Frankfurt, I will hazard a guess you are flying with Star Alliance carriers (United, South African & maybe Luftansa).
Assuming your flights are all connecting without any overnight stops, your bags should be checked all the way through and you should stay airside in the international departure lounges at all intermediate airports. So you should not have to clear customs or immigration anywhere until you reach Livingstone. Nor will you need to claim and recheck your bags.
A lot of connecting flights from the USA to Africa arrive first thing in the morning in Europe and then depart that evening for Africa. In this case, if you wanted to, you could clear immigration and head into town to explore around rather than sitting for 12 hours in the airport, just be sure to get back in plenty of time for departure and be aware that there might be a departure tax imposed (some airports give you a paper certifying that you are connecting not departing to avoid this). Your bags, if checked through, would stay with the airline.
If by chance you have any longer layovers at other stops, then likely you would have to claim your baggage, clear immigration, go find a place to stay for the duration and then check back in.