Could aliens harness stars to keep ahead of expanding universe? In this paper, Dan Hooper of the Fermi National Accelerator Laboratory proposes that advanced aliens might gather stars from other galaxies as a hedge against the dark energy powered expansion of the universe, to provide starlight to power their civilizations. He doesn't suggest a way to do it; he leaves it up to the aliens to figure out how to build Dyson spheres around stars and harness the energy collected thereby to move the stars.  He suggests the possibility of detecting such advanced alien civilizations by the distribution of sizes of stars in galaxies (because only stars in a particular mass range would have long enough lifetimes and low enough masses to move within a useful time).
It's a fanciful and fun idea, but it seems like it might be the least effective way to harness the energy of stars in other galaxies. Wouldn't it be far more efficient for the aliens to use the Dyson spheres to power lasers to direct energy toward their home galaxy? Alternatively, couldn't the aliens power their civilizations (in principle) for a very long time by dropping matter from their own galaxy into the black hole(s) in their galactic center?
I am not asking about engineering methods or choices per se.  I'm asking about the physics that might be behind such choices.  E.g., can more net energy be delivered from a star over a longer time over intergalactic distances by moving the star, or by putting the star's energy into a laser beam and transmitting the light?  And, e.g., how would the energy available within a galaxy via starlight compare with the energy available via dropping matter available in the galaxy into a black hole in the galaxy's center?  In the latter case, it seems that some portion of the galaxy's mass would need to be ejected from the galaxy (or the galaxy would need to expand dramatically) in order to send another portion into the galaxy's black hole. 
 A: Consider converting a galaxy to photons (using your handy supercivilization matter-to-energy converter). Half of them would have to be emitted outwards due to momentum conservation, the other half to your home galaxy where they can be caught and converted to useful stuff. The efficiency will be 50% times the energy loss due to cosmological redshift, $E(\mathrm{arrival})/E(\mathrm{launch})=a(\mathrm{launch})/a(\mathrm{arrival})$, i.e. $$\eta = \frac{1}{2}\frac{a(\mathrm{launch)}}{a(\mathrm{arrival)}}. $$
Now consider turning the galaxy into a photon rocket. Using a fraction $1-f$ of the mass leaving you with a "payload" $mf$ you get velocity $$v=c\frac{1/f^2 -1}{1/f^2 + 1} = c\frac{1 -f^2}{1 + f^2}.$$ 
If we measure distance in comoving coordinates the velocity will be divided by the scale factor $a(t)$, so to travel distance $d$ it will take time t enough that $\int_0^t v/a(t) dt = d$ (this assumes that you keep a constant peculiar velocity in transit). If we use $a(t)=e^{Ht}$ as a caricature of an accelerating universe that gives us $t = -(1/H)\log(1-dH/v)$, which shows there is a horizon we can't beat at distance $v/H$ - beyond this we will never get the matter home. This is true also for lightspeed signals, corresponding to a cosmological event horizon.
Sending the light over distance $d$ will take time $-(1/H)\log(1-dH/c)$ and at this time the scale factor will be $e^{-\log(1-dH/c)}$, giving an overall efficiency of $$\eta=\frac{(1-dH/c)}{2}.$$ This decreases linearly with distance to zero at the horizon. 
Matter sent as a payload fraction $f$ over the same distance will arrive unchanged, but needs to arrive before $t=\infty$. That means the lowest acceptable speed is $v=dH$, giving a maximal fraction, and hence efficiency, $$f=\eta=\sqrt{\frac{1-dH/c}{1+dH/c}}.$$ 
This is more efficient than the energy approach. However, it comes with a great deal of delay.
I briefly sketch a non-photon rocket approach to moving galaxies in section 6.1.4 of my paper on the aestivation hypothesis, based on ejecting hypervelocity stars using gravity assists. This is exceedingly wasteful; I realised later that it might be more useful to see the stars as the payload instead. $v=1000$ km/s is enough to at least overcome typical galaxy cluster escape velocities. 
