In Beyond the Grave Amy and Dan have one really horrible fight over Grace and her memory. Amy begins to think that Grace didn't really care about them and was just using them all those years. Later on, Amy and Dan remember a time they went to a history museum in New York with their grandmother and they all ate pretzels on the stairs. Grace tells them that the museum was nice but this is the best part. The Fridge Brilliance of that is, when Grace says she likes spending time with them more than the museum, she also means her travels and adventures all over the world too (which they didn't know at the time)
Dan getting another picture of his parents after he lost the only picture he had of them in Paris.
Irina Spasky talking about her son's death and how she found her women's polevaulting medal in his pants pocket — he'd carry it around at all times because he loved her so much! and then saving the kids from the fire
In book 9, Storm Warning, Amy and Natalie's short conversation, which doubles as a Tear Jerker. Explanation: The Cahills set a trap for the Kabras in order to steal the wolf fang from Isabel's bracelet. In the process, Isabel is knocked out, and while Natalie tries to help Isabel and Amy attempts to bargain with Natalie for the wolf fang, they have a conversation about mothers.
Jonah's dad in the last book when Isabel kidnaps him along with everyone else's families and ties him up to an explosive. Jonah, who never really knew if his father loved him, sees him singing their childhood lullaby on the screen.
Whenever Amy or Dan run into a relative of their parents, Shep, or Fiske, Amy will see her dad and brother's facial expressions in Dan. Their uncle Shep reminded them so much of their father.
The last book has many of these, but Grace's last messages top this. Especially at the end, when we see the secret end message and its Grace (the audiobook makes it really heartwarming where you hear her talking) giving them The Flying Lemur and her base in Madagascar.
Series 2: Cahills vs. Vespers
In the second series, Evan and Amy's interactions are this a good deal of the time.
And, as usual, any time when Dan and Amy show how much they care about each other (even if they do think of the other as a "dweeb" and a "nerd," respectively).