In my language for both cases below there is only one single verb! I have no idea which word works more appropriately in this sense and if they all work here, then how shall I distinguish these verbs from one another in order to use them more properly:
- Don't let your kid / dog (separate / break away) from you.
- We (separated from each other / broke away from each other) at 4 o'clock and I've not heard from him ever since.
For me, both "break away" and "separate" work, and the only difference between them is that separate is a bit more formal.