What is the process by which a trait present in an ancestral organism is modified by natural selection over time in descendants of that ancestor?

Answer is descent with modification.

Explanation:

Natural selection occurs when an organism which possess favorable heritable traits for survival and reproduction have or leave young ones or offsprings more than its peers, resulting in the increase of the trait in frequency and generation.

Descent with modification as explained by Darwin, is the notion that there will be changes in species over time, and this will produce or give rise to new species, which will be sharing the same ancestor.