One way is if there was a global threat that required the skills of both universes. Let's say an evil wizard from the Harry Potter world teams up with a dark bender from the Last Airbender world. Characters like Katara and Hermione could work together to research a way to stop them. Hermione could use her vast knowledge of magic and Katara her waterbending skills and knowledge of the spirit world. Also, characters like Sokka could provide his strategic thinking to the Order of the Phoenix, while the wizards could offer protection spells for the Avatar and his friends.
In fanfiction, characters can interact in many ways. For instance, if there was a magical object in the Harry Potter world that was related to the Avatar cycle in the Last Airbender world. The characters would be brought together to protect or understand this object. Zuko could visit Hogwarts to learn about the dark arts in an attempt to better understand the dark side within himself. He could meet Severus Snape, and they could have some interesting conversations about redemption and power. At the same time, Ginny Weasley might be interested in learning firebending from Zuko, seeing it as a more physical and exciting form of power compared to her normal wizarding skills.
In a fanfiction, a character like Victor Frankenstein could be transported to Hogwarts. He would be both amazed and confused by the magic there. He might interact with Dumbledore, who could be intrigued by Frankenstein's knowledge of creating life through non - magical means. Meanwhile, the students might be either in awe or afraid of him. The Gryffindor students could try to show him the positive side of magic, while Slytherin students might be more interested in his dark secrets related to creating life.
He might interact as a mentor figure. For instance, if he has some unique knowledge about combat or strategy from his own world, he could share it with Aang to help him in his fight against the Fire Nation.
The characters could interact through a magical portal that connects the two worlds. Harry Potter characters could enter the Demon Slayer world and vice versa. For example, Hermione could use her vast knowledge of magic to help the Demon Slayers understand the more complex aspects of the demons they fight.
They could interact in a school exchange program. For example, Hermione could visit Cackle's Academy and share her knowledge of spells and potions with the students there, while Mildred could come to Hogwarts and show the students her unique way of handling magical mishaps.
They could interact through a magical portal. For instance, Hermione could accidentally open a portal to the Star Wars universe while doing some advanced spell research. Then she might meet Han Solo and together they could start an adventure to find a way to close the portal before it causes chaos in both universes.
Well, consider a scenario where Sonic and his siblings are mistaken for some new kind of magical beings at Hogwarts. The students are curious and a bit scared at first. Ron might make some clumsy attempts to befriend them, which could lead to some comical situations. Meanwhile, Sonic's speed could come in handy during a dangerous situation at Hogwarts, like when a group of Death Eaters attack. And Sonic's siblings could use their music to calm the panicked students and staff, showing a new side of their powers to the Harry Potter characters.
In fanfiction, the characters could interact during a global crisis. Let's say an ancient evil is rising that combines magic and advanced technology. The Justice League realizes they need the help of the wizards from Harry Potter's world. So, they send representatives like Superman and Wonder Woman to Hogwarts. There, they meet Dumbledore and the students. The students are excited to see these super - heroes and together they form a plan to fight the evil. They combine the super - powers of the Justice League with the magic of Hogwarts to defeat the enemy.
Hermione could study the Sword of Truth. She might use her vast knowledge of magic to try and understand its origin and powers. For example, she could search through the Hogwarts library and find some old texts that give hints about the sword.
The characters can interact in various ways. For instance, Harry could teach Lilo some basic spells. Stitch, being curious, might follow Ron around Hogwarts. Hermione could study Stitch's unique physiology. Another way is that the Weasley twins could try to use Stitch in their pranks. And Dumbledore could offer his wisdom to Lilo when she faces difficulties with Stitch.
One common theme is the power of friendship. In both 'Harry Potter' and 'The Last Airbender', the main characters rely on their friends. So in fanfiction, this is often explored further. Another is the battle between good and evil.