My son has made friends with a new boy at school.
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.
There really isn't an etiquette, more or less.