One of the things that happen to your body when you cry is that your whole system works in overdrive. Your heart races and you may sweat before your parasympathetic nervous system kicks in and helps regulate your heart rate, blood pressure, and breathing (via Columbia University). In the process, your body starts to release the endorphins that help calm you.
When you first start to cry, tears get released from the tear ducts in your eyes. They fill the eyes with fluid until the eyes can no longer hold them and the tears drip down your face. Since tear ducts are housed in the lower eyelids, behind the scenes, they are releasing tears that find their way down to your nasal cavity. The result is a drippy, runny nose. According to Healthline, the tears that fall into your nose get mixed with what's in the nose, such as blood or mucus, and then drip down.