Also why, exactly, are they buying the exact assets that they are buying (govt. debt, high-yield bonds, etc..) and why not others (e.g. stocks or put money into startups)? And then, what happens if a debtor pays back its debt? Is that money consequently getting "erased" again (just like it's been created)? What happens if a debtor defaults on its debt? Does that money then just stay in the economy, impossible to drain out? What is the general expectation of the central banks? What percentage of the debt is expected to default and how much is expected to be paid back?
And specifically in the case of central banks buying govt. debt: Are central banks considered "easier" creditors than the public? What would happen if a country defaults on a loan given by a central bank? Would the central bank then go ahead and seize and liquidate assets of the country under a bankruptcy procedure to pay off the debt (like it would be standard procedure for individuals and companies)?
The liquidity injected is supposed to be taken out later, thus removing the inflationary distortion. Whether it will or not is anyone's guess. 2008's injections have yet to be taken out.
Central banks are easier creditors because, while autonomous, they are the same country as the government! So it's technically like owing yourself money. A central bank that cooperates with the debtor country (itself) would never force a default, and is thus never an acute problem. Of course, infinite money printing should lead to dangerous inflation.
The bad news is that by ignoring inequality, they may be just causing it.