The three-body eikonal approach to ion-atom collisions based on the Alt-Grassberger-Sandhas equations is developed. The amplitudes for scattering and charge exchange, written in the impact parameter representation and using the eikonal approximation, are represented as a sum of two terms. One of them describes the scattering via the Coulomb potential acting between the colliding heavy particles. The second one could be factored into a product of two terms. One is an explicitly known factor which contains all the information about the interaction between the heavy particles. The remaining part does not contain any reference to this interaction any longer. For these residual amplitudes effective-two body equations are written down. Explicit expressions for the effective potentials occurring therein, for both the scattering and the transfer channels, are derived in the lowest order approximation. This approach is then used to calculate total and partial cross sections for the electron capture processes in collisions of the ions H⁺, He²⁺ and Li³⁺ with atomic hydrogen. The latter is taken to be in its ground state, except for the case of incident H⁺ when the target hydrogen atom is considered also in some low-lying excited states.