According to general relativity, would gravitational waves experience the same sort of redshift that electromagnetic waves experience due to the expansion of the universe?
And are there astrophysical processes that would be expected to have a characteristic 'frequency signature' analogous to the lyman or balmer series of hydrogen which would enable a measurement gravitational wave redshift.
I'm thinking of processes like baryon accoustic oscillations, certain types of supernovae, neutron star mergers, etc.
Yes, gravitational waves can be redshifted. The answer of this question explains why: https://physics.stackexchange.com/questions/137292/can-gravitational-waves-be-red-shifted The answer to the question above basicly states that doppler effect applies to any wave that propagates at light speed.
Actualy measuring the redshift is much more difficult, because neither redshift nor rest-frame mass can be extracted from the signal. Only the combination of the two can be obtained - M(1+z). This is called the redshifted mass. Deriving actual redshift from redshifted mass is quite complicated as this problem is relatively new and no efficient solution exists yet. This article here describes the derivation of the redshift in detail: http://arxiv.org/pdf/1312.1862v2.pdf From it you can see that it is indeed possible to derive redshift from redshifted mass, but the method is really imprecise as it has a massive 10 - 20% uncertanity.