How to Move the Sun: Stellar Engines

Kurzgesagt proposes moving the entire solar system, so the generations and kilometers question becomes moot: ▶ this video

You might enjoy Kurzgesagt's video on moving the Sun (and with it the whole solar system) they propose a method that would theoretically let us colonize the galaxy in a reasonable (but not trivial) time frame. To your exact point though, since we're moving the whole solar system, everyone would be living their whole lives on the/a ship. ▶ this video

The "time it takes for the passengers of a generation ship to start a new civilization and gather the resources to launch another generation ship" is minuscule compared to the time to make the journey in the first place. Here's a Kurzgesagt video that essentially proposes using our solar system itself as our "generation ship" and getting from one star to the next would take on the order of a hundred thousand years, while the time to set up to travel that way seems reasonable to create in tens of thousands, or even thousands, of years. ▶ this video

It depends on what time scale you're talking about. Here's a Kurzgesagt video that describes how to do it over the course of millions of years: ▶ this video That's a loooong time, obviously, but it's not "theoretically impossible"

Here's the Kurzgesagt video on moving the solar system through the galaxy: ▶ this video I've seen at least one "physicist reacts" video to this that tl;dw's to: "seems reasonable, would obviously require a lot of technological advancement and work"

In case anyone is interested in how grabby aliens (or we!) could travel the galaxy, Kurzgesagt has an excellent video on building a stellar engine to move our solar system at up to 50 light years per million years. Setting aside considerations of whether we'll survive the next few decades as a technological civilization, let alone the next few millions of years, the Caplan engine would let us colonize a significant fraction of the galaxy within a billion years or so. ▶ this video

Saying there is "no evidence" is factually absurd. There are a whole host of possibilities for practical long distance space travel. At very least for small light-weight robots. And it even seems possible that we can viably transport our entire solarsystem: ▶ this video . No evidence indeed... only if you lack imagination.

I wouldn't expect an exhaust plume passing through a wormhole to move the wormhole itself (or the heavily curved "portals" at its ends) any more than a plume would move "flat"/"normal" spacetime. Separately, Kurzgesagt did a video on stellar engines (which can be much less exotic than wormholes); sounds like what you're describing: ▶ this video

Out of every video Kurzgesagt has ever produced, my favorite is this one, which talks about a stellar engine that a type Ⅱ civilization could be using: ▶ this video

And how do we move our spaceship? I was amazed to know that theoretically we can move our solar system. ▶ this video

Extracts: The machine would use a Dyson sphere to collect mass, separate out helium to push the sun, and fire hydrogen at the sun to avoid crashing into the sun. It could push the Sun at a speed of about 50 light years per one million years. Matthew Caplan from Illinois State University designed the machine at the behest of Munich-based YouTube channel Kurzgesagt. The channel then created a stunning video to illustrate the device, which it dubbed the Caplan Thruster, and a paper Caplan wrote describing the machine has also been published in the journal Acta Astronautica. Video links: Kaplan thruster: ▶ this video Dyson sphere: link