Late in a pregnancy your body releases hormones that soften ligaments in preparation for the birth. This can certainly cause back pain in addition to other pain. I had bad hip pain for several years after the birth of my son, but it is much better now.
Also, carrying so much weight in front can cause a lot of stress on your back, especially in late pregnancy. This can affect your muscles and ligaments that hold your back tightly together and can take a long time to heal.
The best things you can do as mentioned is to keep extra weight off, stay physically active, and strengthen your core muscles. Walking is a wonderful exercise for your low back, and if you do it every day for at least 20-30 minutes you will notice results. Also as mentioned, try not carrying your little ones as much as possible also as this can cause or aggravate back pain - strollers are great to help with this. Ibuprofin is good to help keep inflammation down and relieve back pain, but it is best to work with your primary care physician if you need to take it long-term or need to take higher doses.
Warm wishes!