What force accelerates a liquid moving in a narrowing pipe?

You are right. From continuity of the incompressible fluid you have $$A_1 v_1 = A_2 v_2.$$ So obviously the velocity is changing. Thus the fluid is accelerated, and therefore there must be a force causing this acceleration. In this case the force comes from the pressure difference between the wide and the narrow part of the pipe.

_{(image from ResearchGate - Diagram of the Bernoulli principle)}

This can be described by Bernoulli's equation ($p$ is pressure, $\rho$ is density) $$\frac{1}{2}\rho v_1^2 + p_1 = \frac{1}{2}\rho v_2^2 + p_2$$

There is more mass per area behind than ahead of the constriction so since ppressure is force divided by area there develops a pressure difference

The simple answer is pressure.

As you state, from the continuity equation you can see that the velocity $v_2>v_1$. The next step is a momentum balance (like any balance in fluid dynamics: $\frac{d}{dt}=in-out+production$). The momentum flowing into the system is smaller than the momentum flowing out of the system ($\rho A_1 v_1^2 < \rho A_2 v_2^2 = \rho A_1 v_1^2 \frac{A_1}{A_2}$, $\frac{A_1}{A_2}>1$).

The actual force is not the pressure itself, but the pressure difference, or, actually, the difference in force, because on the left the force is $p_1 A_1$ and on the right $p_2 A_2$.

From another conceptual point. Suppose you have a garden hose. Than you have a fixed pressure drop. Now, if you squeeze it, you create a contraction. Part of the pressure drop is now needed to accelerate the fluid at the exit. The overall flowrate decreases, because you also need pressure to overcome frictional forces.