Our reincarnator loses her memories of her home universe Dc/Marvel but has memories of other media. Reincarnator Morgan comes to around 5yrs old. Morgan is the daughter of the Mad Hatter (he doesn't see her as his Alice) though she isn't bad she will be classified as a villain The Red Queen/Queen of Hearts(I'm thinking of Alice Madness Returns) though she becomes a anti hero later.
Morgan has a bare bones system that streams her adventures to everyone that will have a positive relationship (those she will come to see as family, friends or more) with her in her home world only and the system is themed so for example a magic system she only gets magic related powers or items, animal shapeshifter like beast boy she only gets animal genes and gear for a easier time shifting till she gets used to it Alice in wonderland themed she can summon things from all things Alice in wonderland themed 👀. Time doesn't stop in home world or other worlds. If you have her kill depending on the timeline for Dc before justice league the heroes are very against it during or after JL they are more tolerant of it but still see it as last resort, Marvel will be Marvel nuff said. Morgan can find comics of her home universe in other worlds like MHA or other shows. Morgan should have a semi realistic outlook when going to other worlds so she doesn't see the characters as characters but people and takes what she remembers of them with a grain of salt so she has a excuse to get to know them and not be overly familiar.