The general effect, for periodic loading, is to make a photonic crystal, which has a bandgap, i.e., a bandstop characteristic. Transmission is unreliable (with respect to frequency or pulse width or dispersion, say) above the bandgap, and approximates normal transmission line behavior below.
Consider a wave approaching a node: some is reflected back, and somewhat less is transmitted onward. Then at the next node the same, and so on, and all the reflections between them. (The part reflected back to the source gives the apparent change in impedance. Which is only applicable at low frequencies, where the lumped loading looks continuous, which is to say, there is a small phase angle between nodes.)
For a random distribution of some PDF, I wonder if you can solve the same relations, to find the expected value and variance of bandgap center, width (or band edges), attenuation, and Zo. That would be the setup, but I don't know how to proceed from here; it's been a long time since I did statistics, or ensemble transmission lines...
I've certainly not done it with transmission lines, but I'm familiar with the approach used to solve the resistance of non-adjacent points on an infinite resistor mesh: you use the Fourier transform, because you only have to write the indices of the probed locations, and their Kirchhoff equations. Perhaps a similar thing would apply here, but I'm not sure how to bring aperiodic statistics into it; perhaps it would be similar to modeling RF noise.
In general, you will still have the same LF approximation, but it will only be applicable for frequencies less than the largest electrical length between any pair of nodes. Above there, expect peaks and dips as random pairs or groups of nodes act as resonators -- bandpass networks.
Hah, I wonder if you could model single errors as doping in the photonic crystal -- that is, introducing an allowed level within the prohibited band. This wouldn't be so effective for a high density of errors though.
The width of any given "accidental passband", within the stopband, will depend on the ratio of its impedance to the line as usual (either Zo or Z'), and I suppose its insertion loss too.
At some point of refinement, loss will inevitably have to be considered, as insertion losses will be higher than the ideal case, and bands (stop and pass) will be wider.
Oh, there's also the matter of where stopped signals go -- they could transmit all the way through to the far terminator, but be suppressed from any number of intermediate nodes; or they could be reflected to the source. And for the lossy case, they can be absorbed by the network and reach few (or even none?) of the above.
Tim