Transient time in buck converter
This is an inductor current graph, not an output capacitor voltage graph, so it's difficult to tell from it whether the output capacitor discharges through the load (it does).

The current between \$DT_s\$ and \$T_s\$ is not constant, but the derivative of the current is proportional to the voltage difference between the inductor terminals, which during "off" time is the voltage of the output capacitor at this point (not much) plus the forward voltage of the diode (also not much). Compared to the voltage during "on" time, the change in current isn't as pronounced, so it appears flat in this graph (but the \$i_L(T_s)\$ marker is applies only to the time \$T_s\$, not to the interval).