I have done this years ago when my big kids were little. I normally made it a quick run into someone/BF at first. Then he would take us all out to eat another week. Then we would all go on little day trips. My kids only ever meet 3 guy when I was divorce and the last one I married. One of them ever got past the random meeting but he was also our neighbor so that kinda happened by accident. My thoughts are to not introduce kids until you are sure you in a long term relationship but you have no control over your X. My X introduced the kids to his current wife 3 days before marrying her and he has only known her foe 2 weeks.....
I am scared to death how it is going to happen this time. Our divorce is not even done and he has been carrying on with another woman for months. She is already asking to meet our 8 year old. We go to mediation in 2 weeks and I am going to try to get him to agree to not meeting the GF for 3 months after it is final. (we currently live together)