We study the effect of the relative velocity of dark matter and baryonic fluids after the epoch of recombination on the evolution of the first bound objects in the early Universe. Recent work has shown that, although the relative motion of the two fluids is formally a second-order effect in density, it has a dramatic impact on the formation and distribution of the first cosmic structures. Focusing on the gas content, we analyse the effect of relative velocity on the properties of haloes over a wide range of halo masses and redshifts. We accurately calculate the linear evolution of the baryon and dark matter fluctuations, and quantify the resulting effect on haloes based on an analytical formalism that has been carefully checked with simulations in the case with no relative velocity. We estimate the effect on the abundance of early haloes and the gas fraction in them. We find that the relative velocity effect causes several changes: (i) the characteristic mass that divides gas-rich and gas-poor haloes is increased by roughly an order of magnitude, from 2 × 104 to about 2 × 105 M⊙; (ii) this characteristic mass has a large scatter (full width at half-maximum ∼1.5 × 105 M⊙ at z= 20); (iii) the fraction of baryons in star-less minihaloes is suppressed by a factor of 3.3 at z= 20; (iv) the fraction of baryons in haloes that can cool and form stars is suppressed by a factor of 1.5 at z= 20; and (v) there are enhanced spatial variations of these various fractions.