Do top MBAs care a lot about your undergrad school?

So, I'll be coming from a business school from Asia, it's known to be the best in my country, but I don't think nationwide. It's a 4-year program school which specialized in business and I am taking finance for sure. Trying to save up money for my MBA, so taking my undergrad here would help a lot. It's 20k until I graduate compared to 120k for Haas.

I will be working in an investment banking firm in my country for 4 years, then straight to the MBA, I'm aiming high to Wharton or Stern, and Ross, Kellogg, also Booth as subs. ( Even considering Ox-Bridge)

Do they care a lot about where you came from? I mean, Berkeley is the best public school in the world, a degree from Haas I believe would do me a wonder anywhere. Or do they care more about your working experience, GMAT, and your essay?

Need someone to enlighten me, my main goal is the MBA before going to any notable private equity firms. If I need to go to Haas, then that way, just gonna be in a lot of debt after graduating and have to save again for tremendously expensive MBA.