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I'm a D&D fan and Magic: The Gathering fan.

I'd heard that Wizards of the Coast says that Magic: The Gathering planes are Dungeons and Dragons planes. I was so excited and full of idea of adventures.

In my research, I didn't see anything about planeswalkers from MTG.

Is there lore and/or written rule(s) supporting MTG planeswalkers? If so, what/where is it?

The Planeswalkers are mentioned in Guildmaster's Guide to Ravnica

The History of Ravnica section of the book contains the following passage:

Jace is a Planeswalker, with the ability to travel from world to world, and his attention never remains focused on Ravnica for long. Thanks to his involvement with other Planeswalkers, he spends extended periods of time away from Ravnica.

However.... there are no official rules for how Planeswalking works in D&D 5th edition.

The worlds of Magic: the Gathering are not "planes" in the same sense as planes are described in D&D 5e. Instead, they are more similar to the different worlds that D&D campaigns can exist within. Some example of these include the Forgotten Realms, Greyhawk, Eberron, and Dark Sun.

D&D canon

Planeswalking in official canon is likely a form of Spelljamming. In the campaign guide, Spelljammer, all of the worlds of D&D (and their respective Material Planes) exist within crystal spheres that lie within a medium called the phlogiston (not dissimilar from the Blind Eternities in MtG lore). Spelljamming typically requires advanced arcanomechanical vessels called spelljamming ships.

The levels of magic involved in these vessels is far beyond the reach of traditional adventurers until the upper reaches of the Character Advancement table. As such if Planeswalking were ever represented as a character option, it would likely be an Epic Boon of sorts.

Note: There are entities from the Spelljammer setting in official D&D rulebooks such as the Giff found in Mordenkainen's Tome of Foes.

There are no specific rules for Planeswalker characters in D&D, probably because they aren't really needed

If we reduce the MTG Planeswalkers to their key properties; they are really powerful entities/spellcasters and have the specific (and unique in MtG muliverse) ability to travel between planes (there's some difference in the cosmologies, so it's maybe more like traveling between worlds in D&D, but we can ignore that).

D&D already has wonderful ways to represent such characters; high level spellcasters with access to plane shift (and/or gate. That spell is available to most (Cleric, Druid, Sorcerer, Warlock, Wizard) spellcasting classes, and so having that ability be part of the class ability makes it fit neatly into the D&D rules without wonky extra rules or options.

If you wanted a D&D feature/option which could represent the Planeswalker 'Spark' (a MtG term), it would need to be independent of classes and races so it could still represent the variety of Planeswalkers possible, and the ability to cast a 7th level spell, even if restricted to self only or similar, is probably beyond to 'power-scope' of feats and backgrounds and over in the Epic Boon territory (see DMG p. 231). What you do in terms the trauma usually necessary to ignite a persons Spark, is your own business.

As for lore, any lore about MtG Planeswalkers are going to found in MtG specific material, not the 'core' D&D stuff.