Dirac delta convolution with function I've come into a bit of a snag, and thought some more talented mathematicians could maybe help.
I am trying to do the following integral:

$$S(x,t) = \int I(z)\delta(x-G(z,t)) \mathrm{d}z,$$

where $G(z,t)$ is a function which 'pushes' the original function $I(z)$ into $S(x,t)$ at some later time.
I've tried using some Dirac delta identities but have not had much success.
Any help would be very much appreciated. Thank you.
Have you tried to use the decomposition of the "composite" delta function $\delta(f(x))$,

http://en.wikipedia.org/wiki/Dirac_delta_function#Composition_with_a_function

In your case you have
$$\delta(x - G(z,t)) = \sum_{i} \frac{\delta(z-z_i)}{|\partial_z G(z,t)|_{z=z_i}|}$$ 
where the sum goes over the solutions $z_i(x,t)$ of the equation $G(z,t) = x$, so that
$$S(x,t) = \sum_{i} \frac{I(z_i)}{|\partial_z G(z,t)|_{z=z_i}|}$$
Potential problems might arise at points where $\partial_z G(z,t)|_{z=z_i} = 0$.
You'd think of $\int f(x) \delta(x) \; \mathrm{d} x = f(0)$ as picking just the value of $f$ where $\delta$'s argument is zero. I.e., in this case the result is $I(z)$ wherever $G(z, t) = x$.