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 10¹²-$10^{14}\, {\rm M}_\odot$ 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) (σ₈, 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....