Can running without a sports-bra cause serious damage?

Wearing a tight bra since the start of puberty leaves no exercise of the muscles which hold your breasts up naturally. Wearing a flimsy bra after wearing more constricting ones, especially whilst running, could just cause for sudden use of those muscles which haven't been strengthened and are not used to heavy exercise. They could be sore as any unused muscle would be sore from sudden use.
Another possibility is just a natural side effect of the menstruation cycle. A common effect is sore breasts. However, I think the first example is more likely.
I don't believe that what you are explaining is cause for anything serious. Just gradually start getting used to wearing looser bras whilst exercising to let your breast muscles gain strength. Then, they should be perkier and less painful.
P.S: Wearing looser bras or no bra at all for 12 hrs. or more also reduces your risk of breast cancer.

Well, that definitely makes sense! Thanks for the reply. I'll try it out. I never knew had to do with muscles. I always assumed wearing a bra would help fight gravity and keep them perky.

Well most of the time keeping them tight pack can take away their freedom. so better is to remove this but while running don't ignore this this will create hypertension state for your breast.

Anytime a woman runs, jogs and / or walk fast, it is very advisable to wear the proper sport bra but they are not to be worn for normal occasions such as work, home, party, shopping, etc. What happens when you wear a flimsy bra, is the fact that your breasts will want to push downward, pull on the upper part of your breasts and stretch your skin thus making your breasts sag more and at the same time could cause you pain for quite a while. The muscles under your breasts which are between your breasts and your rip cage, are not the ones that hold your breasts. Do not forget that you have mammary glands in your breasts, a lymphatic system in your breasts which is attached to the normal system through your body and you blood circulatory system which is also in your beasts and again connected to the rest of your body system. All these systems create a chain of events which will cause extreme pain if not cared for properly.

If you really want to keep your breasts healthy, use proper garments for different occasion and never wear any of the underwire types of bras as they are known to cause breast cancer.