Let's first talk about how the poor gives the money to the wealthy.... They allocate a huge part of their money (which isn't a lot in the first place) to gambling thinking that this would be the quick fix to their problems. This is how they give the money to the wealthy in this case the casino owners.
I call this as the "cycle of gambling" where the average gambler would fall in an endless cycle of gambling regardless of the result. For example, if a person losses in a game, the natural response from their end would to recover their losses. Similarly when a person wins, they would think that they could win again by gambling their winnings, ultimately losing in the process.
But the money can also go from the wealthy to the poor. If the tax from casinos gets collected by the government and is used effectively, it can help a lot of those in need. The money can be used to improve infrastructure, education, healthcare and etc.
The greatest outlier on what you mentioned is if the money can be used
EFFECTIVELY by the government.
Unfortunately, there are lots of countries in both developing and third-world countries were the government does not efficiently use the taxes for the benefit of their citizens. In this kind of setup, the revenue would be most likely be pocketed by some individuals instead of being allocated for the benefit of the poor.