There could be. It depends on the specific comic universes and story arcs. Sometimes, crossover events or special storylines might include this kind of training scenario.
Naruto could show his Batman fandom by creating his own version of the Bat - symbol on his headband or some part of his ninja gear. It would be a simple and direct way to display his admiration.
Yes, Batman comics have shown Bruce Wayne as a boy on several occasions. These depictions often explore his early traumas and the events that shaped him into the hero we know.
Jason could show care for Batman in fanfiction by having deep, heartfelt conversations with him, revealing a hidden soft side. He might also take on dangerous tasks alone to spare Batman from harm or work behind the scenes to gather crucial information that benefits Batman's endeavors.
The Teen Titans Go version is more comical. The original origin is dark and serious, but in Teen Titans Go, they add funny moments. For example, when Batman starts his training.