But there will be less injection that now right, in case UBI is implemented? No "new" money will appear (unless you suggest printing more and more trillions). This is the same money taken away as taxes. Except now it goes to those in need, instead it will go to everyone: now almost all of that money is consumed and so is indeed a "cash injection", instead the portion of the money - that goes to those already doing OK - will be saved.

Providing a "cash injection" doesn't require the UBI. It only requires taxing the rich. But the rich - yes, they mostly CAN actually leave the country to escape taxation, and operate from elsewhere, they can move their assets - increasingly virtual such as intellectual properly - onto offshore entities too... So there's really no option of taxing the rich.