Controlling classical cardinal characteristics while collapsing cardinals

Given a forcing notion $P$ that forces certain values to several classical cardinal characteristics of the reals, we show how we can compose $P$ with a collapse (of a cardinal $\lambda >\kappa$ to $\kappa$) such that the composition still forces the previous values to these characteristics. We also show how to force distinct values to $\mathfrak{m}$, $\mathfrak{p}$ and $\mathfrak{h}$ and also keeping all the values in Cicho'n’s diagram distint, using the Boolean Ultrapower method of arXiv:1708.03691 . (In arXiv:2006.09826 , the same was done for the newer Cicho'n’s Maximum construction, which avoids large cardinals.)