As per Pablo Hidalgo on Twitter (Usual disclaimer: his Twitter account explicitly says anything he says is "not to be cited as canon", so this is not fully canonically proven yet)
Steele Wars Podcast @SteeleWars Jan 2
Any insight on if #StarKillerBase can redrain the same sun?
Or can it use far away suns from other systems? @pablohidalgo (please/thanks)
Pablo Hidalgo @pablohidalgo @SteeleWars
The Starkiller is mobile. Able to travel distances in hyperspace amid a very populous star cluster.
Looks like WGA script confirms what we see onscreen, to a degree. It does consume its sun. However, what's made a bit unclear in the script is, whether it consumes the entire Sun in one shot. The way the wording seems to me, is that it consumes the sun SLOWLY.
A vast view of the planet -- a MASSIVE SOLARVAC ARRAY
surrounds a port TEN MILES IN DIAMETER.
MILLIONS OF PANELS turn on the ARRAY -- a wave of BRILLIANT
REFLECTIONS. Suddenly, like a planetary-scale TESLA COIL
LINE OF ENERGY, THE POWER OF THE SUN begins to TRAVEL DOWN
to the Starkiller Base planet.
and in Finn's briefing
It uses the power of the sun. As the weapon is charged, the sun is drained until it disappears.
They follow Finn on the snowy hike. On the horizon, THE
LASER SIPHON SHOOTING INTO THE SKY, SLOWLY SUCKING THE SUN DRY.
INT. STARKILLER BASE - CONTROL ROOM - DAY
Technicians at work, the SUN SUCKING seen in the window behind him.
According to the novelization, the energy of the sun is only used to help the weapon get ready - the weapon's offensive energy itself doesn't come from the sun, it comes from Dark Energy.
“As near as I understand it,” Finn continued, “enormous arrays of specially designed collectors use the power of a sun to attract and send dark energy to a containment unit at the core of the planet, where it is held and built up inside that containment unit until the weapon is ready to fire.”
“Impossible,” Ackbar insisted. “Although we know there is more dark energy in the universe than anything else, and that it exists everywhere around us, it is so diffuse that it can barely be detected. Let alone concentrated.”
Finn persisted, despite the discomfort he felt at disagreeing with someone of Ackbar’s rank and experience. “It can be, and it is,” he responded with certainty.
Statura, at least, seemed ready to believe. “If the engineering could be worked out,” he observed, “one would have access to an almost literally infinite source of energy.”
There is no mention of travel capability in the novelization, but it doesn't seem that it's needed - the energy of a star is really really really a lot, in real life - so it would take a lot of time to make a meaningful impact on a star.
To understand this, let's find a random Internet caclulation for Sol:
The Sun emits 3.8 x 10^33 ergs/sec or 3.8 x 10^26 watts of power, an
amount of energy each second equal to 3.8 x 10^26 joules.
In one hour, or 3600 seconds, it produces 1.4 x 10^31 Joules of energy
or 3.8 x 10^23 kilowatt-hours.
Since E = mc^2, in 1 hour it looses (1.37 x 10^37 ergs)/(9 x 10^20) =
1.5 x 10^16 grams or 15 billion metric tons of mass.
It's been doing this for about 4.5 billion years!
Found canon confirmation that Starkiller Base likely couldn't be moved in novelization:
Please note that the following Poe's statement comes AFTER they had seen the schematics of Starkiller base from survey team they sent, which means they would have seen/known if it has a hyperdrive:
“We’d likely get only one shot at it,” Poe put in. “What Admiral Ackbar said about keeping it secret would only work as long as its location remains unknown. Once the First Order realizes that we know where it is, they’d throw everything they’ve got into defending it with ships, mobile stations, and long-range detectors. We might never get close to it again.”
Note that he doesn't simply say "once First Order realizes that we know where it is, they will fore hyperdrive and move it".
This is NOT canon, but my own speculation, but there is one very plausible explanation for why the film visuals make it appear as if a star's actual mass was "slurped into" the weapon in the movie, contrary to what the novelization seems to say:
This was how J.J. Abrams tried to visually depict "Dark Energy" being gathered. Because that is kind of hard to visualize for the filmgoers.