A few quick things.
T Tauri stars: While not well understood, it is generaly believed their energetic behavior has a lot to do with how they are born. Stars form out of large clouds with their own magnetic fields and rotation. When you shrink a big cloud to a little star, it is like a figure skater pulling their arms in a spin, they speed up. The same is true for both the star rotation and the magnetic field (conservation of angular momentum and magnetic flux for the physics minded). Theorists have long known that this was a problem for forming stars at all. Stars need to get rid of this extra spin and magnetic field and we think this is what all this violent early activity is about. It is not clear to me that tossing ten brown dwarfs together will have an effect at all similar.
Restarting our sun: Our sun only uses 10% of its fuel over its lifetime. The problem is all the burning takes place in the core, which fills up with helium (which does not burn in our sun). If we could mix the sun (and probably regularly remix the sun) we might make it last 10 times longer. That, however, might be harder than moving a brown dwarf.
Moving a Brown Dwarf: Hold on, because I’m going throw a little matha t ya. A really powerful solid fuel rocket (say a Proton) can produce about two million Newtons of thrust (a Newton can accelerate 1 kg by 1 meter/second per second.) Ion rockets are 10 times more efficient, so in the distant future we can probably imagine one producing 20 million Newtons for the same price/size. Now lets build one million of them and send them to the nearest Brown Dwarf for a total of 20 trillion Newtons of thrust (2x10^13 N if you like).
Sounds impressive, and it is, but Brown dwarfs are big. At 1/20 the size of our sun, that is 10^29 kilograms (a hundred thousand trillion trillion…from this point on I might have to stay with scientific notation). Newton’s second law says that Force = Mass x Acceleration. Using out million futuristic Proton rockets on our Brown Dwarf gives an acceleration of 2x10^-16 meters per second per second, or 0.2 femtometers per second per second.
But we have a long time right? Say all 20 of our brown dwarfs are spread out over a sphere a light year in radius. To simplifiy things, say we need to move the typical brown dwarf a half light year to get it to the joining spot (it would be slightly longer). That is about 5x10^15 meters. If
our million rockets were firing the entire time, how long would it take it to get there?
100-200 million years. (distance=0.5xaccelxtime^2)
Unlikely any technology could work that long. One could be clever and look for brown dwarfs that were heading in a similar direction, which could certainly cut the necessary time down, but I think you see the problem. I would much rather hook those same rockets to a good sized asteroid and cruise my population to the next star…
But that’s just me.