In your example where the money is literally taken & handed out, it'd result in short-term price surges, and probably have very little re-distributive effect (the money would end up flowing back up rapidly, and flow back down at the same rate as before).
But that's not the only possibility. It should be possible to perform land & capital reform: altering the ownership & organisational structure of companies, causing them to allocate resources differently, causing the prices of goods (and of labour) to change relative to each other, and this can increase equality & overall standards of living. For instance, elite populations tend to hoard land & buildings, leaving them idle to fulfil their higher-level psychological needs (i.e. a big private estate) whereas that land could house a factory that contributes to lowering the prices of goods (or, I'm sure you can think of some other uses). They also tend to order the overproduction of prestige goods, when the effort that went into producing those could have went into producing goods that more effectively increase living standards. The relative price of bread & champagne isn't fixed and when the gap is small, it contributes to inequality, but measuring how much is difficult.
Of course, actually figuring out how to do this is the hard part, but it has been done historically. Usually, the longer it lies, the bolder elites get with misallocations of capital, and the easier it is to come up with good suggestions. You are here (cough, datacenters, cough).