observations about… characters in fiction who have the power to travel into works that are fiction within their world
though this trope occurs pretty evenly spread across different types of source fiction, the approach seems to be different depending on what kind of fiction the character jumps into
Games
(Jumanji, Zathura, Tron… arguably even Alice in Wonderland and Through the Looking Glass, since the former sent her into a world ruled by card games and the latter into a world ruled by chess)
Character enters the work involuntarily, by accident or due to unwise personal decisions, though sometimes facilitated by another character
Video/film/cinema
(example: …Pleasantville?)
characters are sent into the work against their will by another character who appears out of nowhere (need more examples to plot any actual trend)
Books, literature
(examples: Gumby from the claymation TV series; Thursday Next from the novels by Jasper Fforde)
this is the only category in which the character uses the power regularly and voluntarily, to achieve goals.

conclusion
books are the medium of fiction where the audience has the most control (not of the plotline, but of how everything looks and feels, via the power of imagination, as well as the ability to control one’s reading by mere motions of the eyes) ….and this is reflected in depictions thereof.
(conclusion subject to change, however, if I ever find more examples)

Update: more examples have occurred to me

Paintings/drawings/still images

(Examples: Bert from Mary Poppins… some Doctor Who characters I think?)

These, interestingly, are at least as intentional as the storylines about entering the world of a book. I think this tracks with the idea that the audience of a still image brings at least as much interpretation to it as a book’s readership.