  The connection between the nature of a protoplanetary disk and that of a
debris disk is not well understood. Dust evolution, planet formation, and disk
dissipation likely play a role in the processes involved. We aim to reconcile
both manifestations of dusty circumstellar disks through a study of optically
thin Class III disks and how they correlate to younger and older disks. In this
work, we collect literature and ALMA archival millimeter fluxes for 85 disks
(8%) of all Class III disks across nearby star-forming regions. We derive
millimeter-dust masses and compare these with Class II and debris disk samples
in the context of excess infrared luminosity, accretion rate, and age. The mean
dust mass $M_{\text{dust}}$ of Class III disks is $0.29 \pm 0.19~M_{\oplus}$.
We propose a new evolutionary scenario wherein radial drift is very efficient
for non-structured disks during the Class II phase resulting in a rapid
decrease of $M_{\text{dust}}$, whereas disk dissipation is a more gradual
process. We find long infrared protoplanetary disk timescales of ${\sim}$9-10
Myr, which are also consistent with slow disk evolution. Finally, in structured
disks, the presence of dust traps allows for the formation of planetesimal
belts at large radii, such as those observed in debris disks. We propose that
structured disks are thus directly connected to debris disks in one
evolutionary pathway, in contrast to radial drift dominated disks which evolve
to near diskless stars. These results set the scene for a novel view of disk
evolution.
