Question about cutting cacti to force pupping

Hi ,.... this a general question, i know alot of cacti will pup profusely after having the top removed....is this applicible to all cacti? ....and are all cacti able to be started from a calloused cutting as i have tried some which never took like Oreocereus trollii....do the same laws of growth apply to all genera or do some not pup and do some not form roots from cuttings? i have some ugly old plants i would like to remove old bits and start again and also perhaps get pups to start a fresh also ....cheers

Without being the biggest expert in this area, I have always had luck striking from cuttings, or generally plants that have sustained damage to the top have started offshooting around the damaged area.
What plants do you have?
I have an oreocereus doelzianus which I puchased in a clump, with one stem damaged at the tip, which has started to pup around the damaged area now. I have another oreocereus cutting (a roundish clumping one, not 100% sure which it is exactly) which i got at the C&S Society show in November, which took pretty easily. Some rebutias and mamms I have with tip damage have pupped like crazy around the damaged area. The opuntias always behave in the same fashion.
The only really tricky plants I have had are a type of euphorbia which just keeps dying off every time I try to strike from cuttings and the base is always rotting...otherwise I think as a general rule with the hardier plants you can do it

What kind of euphorbia?

blowng, I think that it's pretty standard especially with tips...

I think it's a trigona, it had grown about four feet tall but the spine started to rot, so we cut it into three, and one died off, one is going sort of well and one is struggling. They are quite prone to sunburn too, the 'flat' sort of euphorbias don't strike as easily as the columnar or branching sort I have found.

Yeah i was thinking Oreocereus, Browningia, Pilosocereus and an old solitary growing Mammilaria which are all very old and have ugly scarring and malformed growth patterns from years of mistreatment, but if the same rules off callousing etc.. apply to all cacti then ill give it a go ...i cant stand looking at the history of mistreatment apparent in their growth any longer ...some people just have no heart ...

I thought most Mamms will offshoot without cutting I would think, although I am not 100% sure. I have gotten some quite sick plants over the last couple of months, I just got some in bone dry soil with moss growing on the top Shocking. But so far I haven't lost any and all have responded really well to new good soil and a few feeds.
This is the oreocereus starting to offshoot, it was damaged when I purchased it and I haven't tried to neaten the top up but I was pleasantly suprised to see a new pup