For future surveys of fast radio bursts (FRBs), we clarify information available from cosmic dispersion measures (DMs) through cross-correlation analyses of foreground dark matter haloes (hosting galaxies and galaxy clusters) with their known redshifts. With a halo-model approach, we predict that the cross-correlation with cluster-sized haloes is less affected by the details of gastrophysics, providing robust cosmological information. For less massive haloes, the cross-correlation at angular scales of |${\lt} 10\, \mathrm{arcmin}$| is sensitive to gas expelled from the halo centre due to galactic feedback. Assuming 20 000 FRBs over |$20\,000 \, {\rm deg}^2$| with a localization error being 3 arcmin, we expect that the cross-correlation signal at halo masses of 10^12–|$10^{14}\, {\rm M}_\odot$| can be measured with a level of |${\sim} 1{{\ \rm per\ cent}}$| precision in a redshift range of 0 < z < 1. Such precise measurements enable one to put a 1.5 per cent level constraint on |$\sigma _8\, (\Omega _\mathrm{M}/0.3)^{0.5}$| and a 3 per cent level constraint on (Ω_b/0.049)(h/0.67)(f_e/0.95) (σ_8, Ω_M, Ω_b, h, and f_e are the linear mass variance smoothed at |$8\, h^{-1}\mathrm{\,Mpc}$|⁠, mean mass density, mean baryon density, the present-day Hubble parameter, and fraction of free electrons in cosmic baryons today, respectively), whereas the gas-to-halo mass relation in galaxies and clusters can be constrained with a level of |$10$|–|$20{{\ \rm per\ cent}}$|⁠. Furthermore the cross-correlation analyses can break the degeneracy among Ω_b, h, and f_e, inherent in the DM–redshift relation. Our proposal opens new possibilities for FRB cosmology, while it requires extensive galaxy redshift catalogues and further improvement of the halo model.