convert old full + differentials into new full

Is there a way to convert an old full backup (at time t0) plus a set of
progressing differential backups off of that full backup (at times
t1...tn) into a new full backup (at time tn) by only working with the
backups themselves (not the original sources) and without restoring
anything?

I'd like to do this as a background task ocassionally, so that
newer differential backups, after this background task has
completed, can use the newer full backup as reference.  This way I can
avoid having to do a new full backup from the sources, and I can even
offload the task of building the new full backup from the old one + the
differentials onto a machine that doesn't have access to the original
sources.

Ideally, I'd also like the original differental backups to be converted
into decremental backups from the new full backup so that the old full
backup can be discarded without losing any information.