In general, it is best to put your baby in SNOO awake but drowsy. This helps her learn to self-soothe and fall asleep without being rocked in your arms or at the breast.
However, if your baby is just starting in SNOO and fusses when laid down, calm her fully (swaddle + turn on a strong white noise + feed) and rock her to sleep before gently moving her into SNOO.
Note: It can really help to play a strong white noise in the room as you place your baby into SNOO. That can reduce the chance your baby will notice—and object—to going into SNOO. After 2-6 weeks, she should be ready to put in SNOO drowsy but awake.
Babies naturally outgrow the need for motion by around 5-6 months, and the App’s weaning feature usually makes the transition to the crib quite easy.