Why are equalizers moving average model (FIR)
The reason why almost all linear adaptive equalizers are implemented as FIR filters is that FIR filters are always stable and that there exist relatively simple and effective adaptation algorithms. Note that much work has been done on adaptive IIR filters (e.g., this book by Phillip Regalia), but in practice FIR filters are still the preferred option.

Note that no real-world channel can be ideally equalized by an FIR filter, but that's not as bad as it may seem because an FIR filter can approximate the ideal equalizer reasonably well if its filter length is chosen appropriately and if the adaptation algorithm is tuned to the application. Also, perfect equalization of the channel is not always desired if the channel is not only dispersive (i.e., acts a filter) but if it also introduces noise. In this case a combined criterion is used that takes into account the ISI due to the dispersive channel as well as the degradation due to the noise.