You've come across what's known as the composition formula of integral kernels. And the analogue of multiplication of a matrix by a vector would be the integral transform itself.
These are widely used in, for example, the theory of Green's functions when solving partial differential equations with given initial conditions (heat dissipation, wave equations, Schrödinger's formula and friends etc.)
Note that there are some subtle differences in the properties of a composition defined by your formula and a normal matrix multiplication that can become quite important. It comes with all the warnings usual to Lebesgue integrals. It's also (more often than not) considered with distributions instead of ordinary functions of two variables, because this gives many possibilities the latter would not (like the identity!). I'm not sure from the top of my head whether under some usual assumptions it may happen that the integral does not converge but that would be another factor to consider (at least at the point of making those assumptions).