The "moon" cactus does bloom, but rarely ... they bloom from the colored part, which is a gymnocalycium. The bottom is hylocereus, which does bloom, but i dont know if it blooms while grafted.

The plant is grafted because it lacks chlorophyll, because without chlorophyll it cant perform photosynthesis, which then means it cant produce food, then dies .... so no the babies cant live without being grafted ... which i dont recomend for succulent newbies (i dont know if you are just saying)

Great info! Thanks a lot! Yeah, I was "just saying", I'm not about to start grafting things. :)

Micro, you forgot to say the triangular Hylocereus dies in a year or two.

The burgundy ones and the variegated burgundy ones can be degrafted and do great. I have a variegated burgundy I degrafted last year and it blooms all the time. The new pups that come off the variegated part are albino and the one off the burgundy are variegated.
I tried a green orange and yellow variegated but it never produced any roots.
The Hylocereus base can be used in a larg hanging basket or trained to climp up a pole in a pot. The flowers on those will be huge.