Just as there is such a thing as gowns for nursing mothers, there is also such a thing as breastfeeding activewear. This might seem like a contradiction in terms but really is not. A nursing mother needs a little extra support and protection because of sensitivity, but otherwise can go jogging or working out at the gym. Lactating breasts are not much heavier, they just feel that way. Wear supporting sportswear and underwear that help everything to feel right when you are on the move.
As expected, this type of sportswear is not available in regular retail stores. They are a specialty item and a bit costlier than regular undergarments. They have the extra layers and peripheral support needed to stay as comfortable as possible. Quality like this can be found at a website dedicated to new mothers. One of your best friends really is a product.