In the episode just before the series finale, Carrie calls Miranda and admits she misses Miranda, Charlotte and Samantha and also adds she keeps wondering whether she'd like Paris more if she'd come there with Big. Later on, Big calls Charlotte and asks how Carrie is doing. The three then meet up with Big at a resturant, where Big admits to them he still loves Carrie and wants her back, and asks if he should go to Paris (in particular, saying, "I'll wander the streets until I find her") or leave her alone to be with Aleksandr. Charlotte and Samantha both look at Miranda, who simply leans forward and says, "Go get our girl."
Harry's proposal to Charlotte. Especially since it looked at first like he was going to reject her.
Charlotte and Harry's interactions seem to be almost made of this.
In particular, when Charlotte puts up her last Christmas tree (in July) knowing it's a tradition she will give up due to her conversion to Judaism. She points out a particular favorite ornament, and Harry insists she hold on to it because of how much it means to her. Easily one of the sweetest moments in the whole series.
Any time Miranda displays that she does in fact have a heart (particularly when she rescues Steve's mother and cleans her off afterward)
In the series finale, Smith comes back from a movie shoot simply to tell Samantha he loves her. Her response: "You have meant more to me than any man I've ever known." Which, for her, is saying something.
A less traditional but still quite touching Heartwarming Moment occurs in the movie when Carrie comforts Charlotte, who is scared that something bad is going to happen to her because her friends have had such a bad year and she herself is getting everything she ever wanted.
Carrie: Honey, you shit your pants this year. Maybe you're done.
A meta-example. In the season six premiere Carrie runs into Aidan and his baby son Tate. As she approaches Tate has a really happy and excited reaction. That was actually Sarah Jessica Parker's own son, happy to see his mother coming up to him.
Carrie immediately coming to comfort Charlotte when she hears about her miscarriage.
Near the season finale, Anthony drops his abrasive attitude to ask Charlotte if he'll still be an important part of her life, now she has a baby. Charlotte hugs him and replies that Anthony will always be important to her, and he responds with a genuine smile.
Aiden's proposal to Carrie, and his laugh of surprise and happiness when she says yes. Harsher in Hindsight after she later broke up with him, but still...
Another sweet, but very small, moment occurs between Charlotte and Anthony in season 6. Charlotte mentions that she was Prom Queen in high school, and he responds with a proud, 'Of course you were, darling.'
When Miranda calls Carrie and tells her she's gone into labour, Carrie is in a carriage with Big. When she freaks out about not meeting Miranda in time, Big promptly pays the cabbie four-hundred dollars to take them to the hospital.