Not sure if this little philosophical discussion is worth a whole thread, but consider this:

1.) Liches exist in Magic.

2.) The popular cultural touchstone of the lich is that they put their soul into various items so that they cannot truly die unless those items are destroyed.

3.) WotC implicitly accepts this trope via various lich-themed items. They also explicitly accept this trope via cards like Phylactery Lich, that call out phylacteries plainly.

With this in mind two options come to mind:

1.) A powerful lich could, in theory, give their phylactery to a planeswalker, allowing them to "respawn" on whatever new plane the 'walker leaves it on when the lich's body is destroyed on their current plane.

2.) Assuming some metaphysical occurrence prevents the lich from respawning elsewhere, they are stuck on their home plane but are immortal and/or they are stuck in the respawn cycle forever, effectively lost in an endless purgatory.

Not all lichs in Magic work like that, though. Jarad and Svogthir just preserved their soul (or at least their mind?) in their own bodies, and Geth's head outright states he isn't anchored to Memnarch's soultraps because he doesn't have a soul.

That said, I wouldn't mind seing the phylactery bit explored in the story. One thing I could imagine is a planeswalker carrying around a lich's phylactery, which might then make it easier to summon that lich permanently (also to other planes) because they might be anchored to the place where the phylactery is, not the place they were summoned from. Undead travel companion, yay!

I feel like it would be easier to just use a green creature tutor spell or something.

Long story short as near as we can tell:
No.
Magic Liches don't seem to respawn. They just... never die.

So can you just try to pull one with you to another plane and have it survive?

Well, the body wouldn't make it through the eternities, but technically the lich wouldn't die.
There just wouldn't be anything left of it except the phylactery.

The phylactery contains their soul, yes?

So a walker could just grab a body and bind it to the phylactery whenever they arrive somewhere.

I like this idea.

Liliana proved that something similar could be done with spirits in AoA, but one of the main distinctions of a lich vs a zombie is autonomy... and if a second party binds a soul, that's essentially trading a phylactery for a phylactery, and is done outside the undead's control.

Using the D&D 5th edition rules, I planned to make a single mono-black planeswalker do EXACTLY that. The timing is uncanny after I posted on D&D Beyond forums about this last week. At first I imagined a spell (albeit likely a blue spell) could be learned to teleport to the phyalactery/phylacteries, but the "respawn very close to a phylactery" could work fine if the lich-walker in question put time and effort to craft the phylactery in such a way to do so.

My take on the character is a Black Artificer from Kaladesh fearing his short-lived-ness by considering the Aetherborn's short lifespan in his own human timespan ("they don't live long...but neither do I!") and upon planeswalking to say, Innistrad, learns to craft a phylactery or 2, (or more as time goes on) and relying on other unassuming planeswalkers to spread them to other planes in order to abuse the cycle of regeneration.Of course using D&D logic.... designing them to belong on the planes they may be taken to, and upon resurrecting on that plane, adding in failsafes to attract souls (Spirits) to fuel the phylacteries.

The timing is not uncanny. The post in the Beyond forums prompted my original post.

Oh, that's also a likely enough explanation