Either a set of humongous pumps, or a humongous number of smaller pumps working in vertical relays.

Drainage is not as big a problem, as long as the pipes are big enough. Since flow is intermittent, a "slug" of water (plus whatever) pushes a column of air ahead of it and pulls a vacuum behind. In a system operating at one atmosphere that uses gravity as the motive force (i.e., no pumps), pressure equalization is accomplished by sizing the piping to be no more than half-full... and they will be, to comply with code. Additionally, one plans for horizontal offsets of the vertical mains (called 'stacks' in the trade) every few floors to reduce the velocity of falling... stuff.

The only time a gravity system will 'back up' is if it's clogged - then, whichever is the new lowest point of exit will begin to receive the discharge. Please don't flush disposable nappies or feminine hygeine items down the loo.