My son has made friends with a new boy at school.
There really isn't an etiquette, more or less.
One thing that worked for us was to trade off play dates right after school. I would pick up the kids from school and take them back to my house and next time she would pick up both kids. the first time we did this I think I said something along the lines of why don't I pick up both kids right after school and then I can bring A back to the school when its time to pick up the older kids. That way they will get a couple of hours playing alone and you don't have to come up to the school twice. The mom gave a note to the teacher saying A was coming home with me that day, so the teacher released both kids to me at pick up.
With my son (who is older) it was a little harder to start the solitary playdates. Part of that was because he was friends with a group of kids who were the older siblings so the moms were a bit nervous about starting solitary playdates. We actually started around the winter holidays as a way to get some holiday shopping done without the kids along. After that first time it became easier.
I think it might be easier to suggest something along the lines of why don't we have x over on Thursday right after school and you can come get him at 4pm or if it would be easier for you we can drop him off.