If you go visit the Vatican Museums, a ton of the art on display there (possibly most of it) is pre Christian, specifically of Roman origin. Why did they put it at one of the centers of Christendom, instead of either leaving it in the Forum or destroying it outright like they did with others (such as the statues that were on the Pantheon)?
What we see in the Vatican Museum is but a small fraction of the artwork that adorned Rome in its heyday. Much of it was looted, destroyed or melted down in the Middle Ages. The gold artwork, obviously, was the first to go. The innumerable bronze statues were melted down to make bells or cannon. The even more innumerable marble statues were burned to create lime mortar for construction. Many architectural elements were either incorporated into new structures or moved to new locations and used as random building materials, or burned for mortar. Very little public artwork survived; the famous bronze equestrian statue of Marcus Aurelius was spared only because it was mis-identified as being of Constantine the Great, the first Christian emperor. Most of the ancient artwork in the Vatican Museum is the result of later excavations, when wealthy and cultured Renaissance nobles took an interest in ancient art again. Their names are often attached to the most famous works: Farnese, Aldobrandini, Ludovisi etc..
I think that individual objects were moved for specific reasons at singular moments in time, and that there was not a coherent program by the Papacy to remove art from the Forum to the Vatican. Could you provide an example of what sculptures you're interested in finding out about?