Running shoes are worn worldwide and so prevalent that people wearing them are no longer marked as "gringoes", so why worry about that. Keep your old pairs, wear them on the outward bound segments (unless you're travelling with associates wearing business wear), use the shoes on the trip, and toss them away when you no longer need them. If you have to pack the shoes in your suitcase, stuff the shoes with socks, toiletries, briefs, even a small camera fits in there and is well protected, eyeglasses in a case, etc. Or you could mail the shoes ahead to your destination.