Van der Waals heterostructures consisting of graphene and transition metal dichalcogenides (TMDCs) have recently shown great promise for high-performance optoelectronic applications. However, an in-depth understanding of the critical processes for device operation, namely interfacial charge transfer (CT) and recombination, has so far remained elusive. Here, we investigate these processes in graphene-WS$_2$ heterostructures, by complementarily probing the ultrafast terahertz photoconductivity in graphene and the transient absorption dynamics in WS$_2$ following photoexcitation. We find that CT across graphene-WS$_2$ interfaces occurs via photo-thermionic emission for sub-A-exciton excitation, and direct hole transfer from WS$_2$ to the valence band of graphene for above-A-exciton excitation. Remarkably, we observe that separated charges in the heterostructure following CT live extremely long: beyond 1 ns, in contrast to ~1 ps charge separation reported in previous studies. This leads to efficient photogating of graphene. These findings provide relevant insights to optimize further the performance of optoelectronic devices, in particular photodetection.