Would the other Yonko go to Marineford to save one of their own?
For the purpose of saving one of his own? No.
As we see in Marineford though, he will definitely try to come if it's to be a part of some large globe-spanning conflict.
Let's assume YAMATO is captured and about to be executed, would he go?
I'm rather certain that Kaido hardly cares about Yamato at all, and I suspect that if Yamato ran off on her own like Ace did, and got themself captured, Kaido wouldn't bother to save them under the pretense of "You brought this upon yourself for disobeying me". That's kind of the impression you get from a dad who shackles their child with an explosive collar.
Then again, if the narrative was that Yamato getting captured and Kaido not responding made Kaido look weak, I think he'd show up to try and wreck Marineford rather quickly.
Would Shanks go if it was Yasopp,Roo or Beckman?
Neither Yasopp, Roo or Beckman would be put into that position except over Shank's dead body, I suspect, given that Shanks basically an older Luffy. So the situation wouldn't happen, but if it did, he would probably show up. Then again, maybe not, since Shanks tends to be really cautious and calculating and attacking Marineford as a single Yonko isn't something which can be done in a way to likely succeed.
Would Bigmom go if it was any of her children?
Ha!
Of course not!
She doesn't give a sh*t about any of them.
Big Mom only fights in conflicts if she's sure that she can win. She isn't sticking her neck out for anybody.
