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 the 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 spellcasting classes (Cleric, Druid, Sorcerer, Warlock, Wizard), 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.
The Plane Shift: Amonkhet article specifically recommends doing this in its Planeswalkers and the Multiverse appendix, i.e. playing high level characters and using the plane shift spell to emulate planeswalking. However, it does not provide any specific rules for doing so. The closest is giving each character the ability to cast plane shift (self-only) under some (vague) constriction. It also advises that you talk with your players about them following up on plot hooks and not just going wandering about the multiverse. (Thanks nick012000 for pointing this out.)
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 the 'power-scope' of feats and backgrounds, and over in the territory of Epic Boons (see DMG p. 231) instead. In fact there is a specific Epic Boon which does pretty much that:
Boon of Planar Travel
When you gain this boon, choose a plane of existence other than the Material Plane. You can now use an action to cast the plane shift spell (no spell slot or components required), targeting yourself only, and travel to the chosen plane, or from that plane back to the Material Plane. Once you use this boon, you can’t use it again until you finish a short rest.
While these Boons are intended as progression/advancement after level 20, there is nothing to stop you from granting them earlier, nor from simply giving them at character generation if you would like them to already be planeswalkers.
As for igniting the Spark, PS:A does suggest an optional rule that:
a Planeswalker who is about to drop to 0 hit points can make a Charisma saving throw with a DC equal to the damage taken. On a successful save, the character instead takes no damage and planeswalks away. It’s up to the DM what plane the character ends up on, because this isn’t usually an intentional process.
You can probably extend this rule to other traumatic events, with an appropriate DC (if desired). As a general note, be careful with your table's boundaries if you're planning to expose them to trauma.
As for lore, any lore about MtG Planeswalkers are going to be found in MtG-specific material, not the 'core' D&D stuff.