The relationship between Harry and Draco in this fanfiction would be greatly affected. Since Harry has a connection to Draco's mom, Harry could become more empathetic towards Draco. He may realize that Draco is not just the bully he once thought. Draco might struggle with the idea that Harry has this special connection to his mom. However, as the story unfolds, they could learn to communicate better. Harry could share insights about Draco's mom that Draco never knew, and this could bridge the gap between them, leading to a less hostile and more nuanced relationship.
Well, perhaps Harry uses his quick thinking and his knowledge of magical creatures. He might step in just in time, using a spell to distract Buckbeak before it can harm Draco. For example, he could use a simple levitation spell to lift a small object and draw Buckbeak's attention away from Draco.
In this fanfiction, Harry might also save Draco by appealing to Lucius' sense of self - preservation or family. He could threaten to expose some of Lucius' secrets or illegal activities if he doesn't let Draco go. Or, if there is some underlying family drama involved, Harry could use that to his advantage. For instance, if there's a feud within the Malfoy family that Harry is aware of, he could manipulate the situation in a way that makes Lucius back off and leave Draco unharmed.
Well, in this kind of fanfiction, Draco and Blaise saving Harry and leading to a romance could be quite a wild ride. It might involve some magical adventure where they team up against a dark force. The change in their relationship with Harry could be gradual. At first, they might be reluctant allies, but as they face danger together, they start to understand each other on a deeper level. The romance could be a result of shared traumas and the realization that they have more in common than they thought, and it's a fresh take on the relationships within the Harry Potter universe.
In this fanfiction, a plotline could involve Harry being magically bound to Draco's mom in some way. As a result, he has to stay close to Draco to protect the bond. This could lead to many comical and tense situations between Harry and Draco as they navigate their new relationship. Another possibility is that Draco's mom has left behind a powerful artifact, and Harry is the only one who can find it. Draco, thinking Harry has ulterior motives, tries to stop him at first but then realizes they need to work together. Also, there could be a plotline where Harry accidentally uses a spell that links his mind to Draco's mom, and he has to figure out how to break the link while dealing with Draco's reactions.
Well, it could be that Draco has a change of heart early on. He realizes that the Dursleys are truly awful to Harry. So he uses magic to create distractions. He could make strange things happen around the house that scare the Dursleys so much that they just want Harry to leave. Then Draco could be waiting outside and whisk Harry away to a safer place, like a hidden place in the wizarding world where he has set up a little haven for Harry.
Maybe Draco discovers a hidden weakness of Voldemort. Draco, having been raised in a pure - blood family, might have access to some ancient magic knowledge. He uses this knowledge to create a situation where Voldemort is temporarily incapacitated, giving Harry the chance to escape or defeat Voldemort. For example, he could find a way to disrupt Voldemort's connection to his Horcruxes for a short time.
Harry may save Draco from Buckbeak simply because he is a good person at heart. He has always been against unnecessary harm, and seeing Draco in danger from Buckbeak, his instincts would be to help. It could also be that the author wants to explore a different side of their relationship, where Harry's compassion overcomes their rivalry, and this event becomes a turning point in how they interact with each other in the fanfic world.
Fanfiction.net could also be a place to search. However, the tagging system there might not be as refined as AO3. But if you patiently browse through the Harry Potter section and keep an eye out for the key elements of Blaise and Draco saving an abused Harry in the summaries or titles, you may come across some relevant stories.
In some fanfictions, it might be due to a prophecy or some magical bond. For instance, there could be a prophecy that Draco's life is intertwined with Harry's in a way that if Draco dies, something bad will happen to Harry or the wizarding world. So Harry takes on this overprotective role to ensure Draco's safety. It also adds an interesting layer to their relationship, changing it from the typical rivalry in the original books.