Saving those baby tulips may not be worth the effort. The daughter bulbs will not likely develop into plants as beautiful as the parent. In most regions, tulips decline in vigor over time rather than increase in size. Tulips (except for species varieties) simply don’t last forever – 2-3 years of blooms is all you can really expect.
If you want long-lived bulbs, choose species tulips when you replant. However, if you live where tulips grow exceptionally well, and you are a bit adventurous, you could replant the daughter bulbs and see what you get. Space the bulbs as you would full-size ones. It will likely take several years for them to reach blooming size.