The classic expander cycle engine, the RL-10 starts as follows:
The RL-10 engine starts by using the pressure difference between the
fuel tank and the nozzle exit (upper atmospheric pressure), and the
ambient heat stored in the metal of the cooling jacket walls. The
engine “bootstraps” to full-thrust within two seconds after ignition.

A typical plot of the valve movement during engine start is shown in
Figure 2. To initiate start, the FSOV is opened and the fuel-pump
discharge cool-down valve (FCV-2) is closed.The interstage cool-down
valve (FCV-1) remains partially open in order to avoid stalling of
the fuel pump during engine acceleration. The pressure drop between
the fuel inlet and the combustion chamber drives fuel through the
cooling jacket picking up heat from the warm metal. This pressure
difference also drives the heated fluid through the turbine, starting
rotation of the pumps, which drive more propellant into the system. At
start, the OCV also closes partially, restricting the flow of oxygen
into the combustion chamber. This is done to limit chamber pressure
and ensure a forward pressure difference across the fuel turbine after
ignition of the thrust chamber.
As the turbopumps accelerate, engine
pneumatic pressure is used to close the interstage cool-down valve
completely and open the OCV at pre-set fuel and LOX pump discharge
pressures. The OCV typically opens very quickly and the resultant
flood of oxygen into the combustion chamber causes a sharp increase in
system pressures. During this period of fast pressure rise, the thrust
control valve (TCV) is opened, regulated by a pneumatic lead-lag
circuit to control thrust over-shoot. The engine then settles to its
normal steady-state operating point.

Source: TRANSIENT SIMULATION OF THE RL-10A-3-3A ROCKET ENGINE