Usually, the resumptive modifier works as follows:
Life is all about moments, moments that record achievements, moments that record time with family.
I wonder what is the grammatical function of the italicized parts. Apposition?
The more important question is if the following grammatical?
Life is all about moments, some moments that record achievements, but many more moments that record time with family.
Comment: The "function" of the italicized words is simply that of words in general when used to construct sentences. Perhaps what you should be asking about is the function of the deleted words whereby three statements have been collapsed into a single utterance with repeated elements discarded: Life is all about moments. Life is all about moments that record achievements. Life is all about moments that record time with family. As a stylistic alternative to Life is all about moments that record achievements and time with family.