Do you know of effective and simple methods to compensate for Doppler due to movement for the CDMA protocol?
I sure wish I had a universal method for that that is any smarter, but:
Unless you have enough SNR and computational power so that you can treat the chips as symbols of a higher rate single-carrier system and do the usual frequency correction as if it was just a normal single carrier system (e.g. higher-order phase control loops), you'll have to brute force your way through frequency hypotheses.
That's among the reasons why high-bandwidth communications switched away from CDMA (as used in UMTS and 802.11b, for example); without very specific restrictions on the code, the best you can do is decoding a set of frequency-shifted versions of your signal. One of them will work best – the one with the correct frequency offset.
Thinking of systems, not all is lost – one could have a non-CDMA preamble where you do an initial frequency error estimate, and then periodically insert pilot symbols in your CDMA data payload to keep track of phase drift. This really happens in a very narrow window between channel coherency and desirable maximum spreading gain.