  The population of fast radio bursts (FRBs) will continue to diverge into two
groups depending on their method of discovery: those which can be localised,
and those which cannot. Events potentially less useful for astronomical and
cosmological purposes due to limited localisation will accumulate with the
advent of new facilities and continued efforts by, e.g., the SUPERB
collaboration, which may require afterglows or multi-wavelength counterparts
for sub-arcsecond localisation. It is important to exploit these sources to
their maximum scientific potential. We perform analysis of FRB dispersion
measures (DMs), considering different theoretical FRB progenitors with view to
place more rigorous constraints on FRB redshifts, in particular for large
statistical samples, via their DMs. We review FRB DM components, and build
redshift-scalable probability distributions corresponding to different
progenitor scenarios. We combine these components into a framework for
obtaining FRB DM probabilities given their redshifts. Taking into account
different possibilities for the evolution of progenitors across cosmic time we
invert this model, thus deriving redshift constraints. Effects of varying FRB
progenitor models are illustrated. While, as expected, host galaxy DM
contributions become decreasingly important with increasing redshift, for
AGN-like progenitor scenarios they could remain significant out to redshift 3.
Constraints are placed on redshifts of catalogued FRBs with various models and
increasingly realistic models may be employed as general understanding of FRBs
improves. For localised FRBs, we highlight future prospects for disentangling
host and intergalactic medium DM components using their respective redshift
scaling. We identify a use for large samples of unlocalised FRBs resulting from
upcoming flux-limited surveys, such as with CHIME, in mapping out the Milky Way
contribution to the DM.
