Two moments that stand out for this trope are when Mei first meets Totoro and falls asleep on its chest, and when Totoro finally realizes what the umbrella does.
How Totoro continues holding the umbrella over his head even when he steps into the Catbus? That was pretty adorable, not gonna lie. The next time they see him is on a balmy summer night — and he's still holding it.
The Catbus' destination locator and subsequent meow "MEEEEI" is a definite "Aww" moment. FOREST HILLS > CEMETERY/SHRINE > MEI
Right after the Catbus helps Satsuki find Mei, it switches its sign to Shichikokoyama Hospital, offering to transport the girls to their mother.
The fact that the largest Totoro seemed absolutely delighted to be of assistance in searching for Mei, not minding a bit that he was awoken from his nap by a frightened human girl pleading for his help.