In these cases, rich people will buy as much BTC as they can and took them out of circulation.
I think you are assuming the value will continue to increase, but it won't with full adoption. Rich people will have little incentive to hold bitcoins over other assets, just as they currently have little incentive to hold dollars.
Poor will have little chance to buy any BTC. When most of the BTC will be out of circulation price of it will go out of reach of normal people. The whole world will be in an economic crisis and violation of the law will be common because people will not have any BTC to buy food.
Why? Let's say that 99% of the bitcoins are removed from circulation by rich people. That still leaves 21 trillion satoshis for everyone else to use. There would be no shortage, and people would still be paid in satoshis that they can use to buy food.