It's not necessarily the case that there is any kind of equivalence. After all, the 6/8 tune could be a jaunty jig, while the 4/4 coould be a slow air.

I must say, to go from 6/8 to 4/4 without a pause, even a slight one, seems a bit unnatural.

If the score doesn't specify that e.g. 'dotted crochet = crochet' (using the symbols for these), then it's up to the musos to make it work based on what feels right.