It is my believe that in a free market where we all play by the same rules poor people will naturally gain more wealth relativily compared to rich people.

Unfortunately, that's not how it works. Read this study for example.

As for basic income, I'm not sure why you're assuming it's global. Universal basic income will most likely happen nation by nation just as universal health care happens nation by nation, not globally.

Also, your inflation assumptions are flawed. There are far too many variables in the equation to just assume massive inflation. Please read this article to get into the weeds on that one.