There are a few tricks, in my experience, that make this possible.
Simple Case
In the summertime, there is very little need for heavy clothing. If it is warm one can board a plane, train or bus wearing just shorts, shirt, underwear, a hat, and sandals. Basically, just bring nothing besides wallet, passport, and toothbrush.
While it is possible to re-wear your clothing day after day, most people find this a bit gross. So clothing should be washed at the end of each day! Its easy - just start taking a shower with all your cloths on, and wash & rinse each piece of clothing with body wash or soap before you taking it off. Hang the clothing to dry while you are sleeping, and you'll have fresh clothing in the morning.
Complicated Case
What if you will go to the beach, and you don't want to swim in your shorts? What if the place you are going is too humid, and your shirt won't dry overnight? What if you are staying in a youth hostel, and thus can't sleep naked?
In this case, wear the bathing suit and any other spare clothing onto the airplane. Bring an empty plastic grocery bag in your pocket. Once on the final leg of your journey, you can stick the excess clothing into the plastic bag.
Very Easy Case
If you are going to visit someone, just make plans to borrow their clothing while you are there.