We saw that happen in many cases in the recent period of austerity, because scarcity of resources is a powerful argument to convince the rich, then the middle class, to renounce their benefits in favor of the poor.
As for "there has not been UBI" I disagree. Many supporters provide examples such as the Alaska dividend. There are many lessons to be drawn from many social policies in developed countries.
In France alone we had two BI-like programs which, for various reasons, underwent slow financial and political destruction: - the RMI/RSA, about which I wrote in the article - less known is the "régime des intermittents du spectacle", which works like a universal unemployment insurance for freelancers in the entertainment industry (and it costs so much money to the taxpayer that governments regularly try to kill it). It only survives because the entertainment industry is a powerful lobby in France.
There has not been UBI so history tells you little. By supplement the market you mean