One health risk from going through multiple pregnancies is the risk of a uterine prolapse, also called a prolapsed uterus. The weakening of ligaments basically leads to the cervix and uterus "sagging". In the most severe cases, the uterus ends up in the vagina.
Parity and obesity were strongly associated with increased risk for uterine prolapse, cystocele, and rectocele.
Pelvic organ prolapse in the women's health initiative: Gravity and gravidity
Parity in this context is the number of pregnancies that led to a viable gestational age (pregnancies excluding miscarriages)
Uterine prolapse makes subsequent pregnancies high-risk and can lead to fetal and maternal problems.
The general risk of anything happening increases due to going through multiple pregnancies in a short time span, though. Without sufficient recovery between pregnancies, women are at a higher risk of placental abruptions and placenta previa. For the baby, this leads to a higher risk of premature birth with the associated issues of low birth weight and small size.