My route is london-beijing-kunming-vientiane but I'm not leaving the airport in China. Do I qualify?

Sara, your route is eligible for 24 hour visa free transit, but under the condition that your entire transit time in China (two layovers and a flight between Beijing and Kunming) is less than 24 hours. In other words, the transit time is counted from your first scheduled arrival in China, in your case in Beijing, until your final scheduled departure from China, in your case from Kunming. So, if your entire transit time is less than 24 hours, you don't need any kind of visa.

You will print out your flight tickets and have those printouts with you. Also, you will declare at check-in in London that you will travel to Vientiane via China using 24 hour visa free transit. After a short inspection of your tickets, you will be allowed to board the plane. You will inform yourself at check-in about the luggage policy because, although you are probably traveling on a single ticket, it may be the case that you will have to collect your luggage at some point and perform a recheck to continue your journey to Vientiane.

Upon your arrival in Beijing, you will immediately approach 24 hour visa free transit counter where you will apply for a temporary entry permit to be allowed to continue your travel toward Laos. Also, you will clear immigration in Beijing. After that, you will board for your flight to Kunming and from there to Vientiane.