In and of itself, the presence of multiple frequencies does not produce any distortion or frequency modulation. The idea of the high frequency waves "riding on" (or being "pushed by") the low frequency waves can be misleading in this context. Intermodulation effects require some non-linearity in the system, i.e. a distortion model.
Most the most basic distortion models only non-linearly transform the amplitude of the signal; this will not model the effect that you've identified.
A model that includes a nonlinear function of both the amplitude and the time-derivative of the amplitude is required to model the effect.
I do not know of any specific models that incorporate this feature, but in terms of the modeling details, the feature that your are looking for is a
distortion model that includes the derivative of the signal as one of its inputs.
ROM of the Effect
Let's just assume that we have a tweeter that is affixed to the cone of a low frequency driver as is envisioned in the question (Note: I'm not sure that this is a good model for single-driver speakers...). I'll assume a 1mm (=0.001m)
amplitude for the low-frequency driver's amplitude. In order to get a 0.5% (1.005 factor, just less than 10 cents) Doppler shift, the frequency of the "low" frequency driver would need to be about 330Hz. This is in the realm of plausibility and perceptability.
A basic (no-memory) non-linear transformation can be represented as
out(t) = a0 + a1*in(t)+ a2*[in(t)]^2+a3[in(t)]^3...
If we assume that there are just two frequencies, f1 and f2, you can see that
out(t) = a0
+ a2*[f1(t)+f2(t)]^2 = a2*[ f1^2 + f2^2 + 2*f1*f2 ]
it's in the quadratic, and higher order terms, of the nonlinear transfer function that produce the frequency modulation. This type of non-linear amplitude transformation is what is most commonly used in distortion effects simulations.
One aspect of the speaker problem that is less commonly modeled is an explict dependence of the output on the derivative of the input signal:
out(t) = F( in(t), in'(t) )
(in'(t) is the time-derivative of the input signal)
Note that for single-frequency signals the the amplitude of the derivative
of the signal is scaled by the frequency itself -- this is an intrinsic high-frequency boost in the derivative signal.
As with the simple model, one can Taylor expand the transfer function to get the intermodulation products; but this will be systematically different from the simple (amplitude only) non-linear model above due to the fact that the time-derivatives are scaled by the frequency.