The first thing to say here is that your children’s thoughts and reactions to a new partner are going to be different from yours.
This is their mum or dad moving on, not their partner.
Age plays a big part in this process. Often younger children will adapt more quickly than say teens, but every child is unique and it’s very much dependent on the circumstances surrounding the initial separation as to how they might present a reaction towards a new partner.
Although there are no hard and fast rules, it is a good idea only to introduce a new partner when you are as sure as you can be that the relationship is becoming more committed.
A gradual introduction on neutral territory for a short period of time initially and then building up to more contact is often a good way of introducing children.