Flat tire rule is an unwritten rule exclusive to US carriers that might allow you to show up late and miss your flight for reasons beyond your control such as flat tires/traffic and get a free rebooking to a later flight (no change fee, no difference in fares, no cancellation of future legs). Because this is an unwritten rule, you are at the mercy of the carrier as to whether your situation is worthy of the flat tire rule. The agent at the airport might tell you to pound sand. To apply the flat tire rule, you need to be at the origin airport in-person. You cannot do this by phone. After 2 hours late, carriers won't listen to your excuses and all remaining legs of your ticket are gone.
With non-US carriers, there is no flat tire rule at all. You miss your flight regardless of any good excuses, you forfeit all remaining legs of your ticket.
Because I agree with others than 90% of the time this will work out, I'm not suggesting what to do. Only you can decide what to do. It'll probably work out. But if it doesn't, don't expect an ounce of sympathy. I've flown enough to see people try this and get screwed by a one-hour WN flight on a sunny day from the Bay Area to the Los Angeles area.