A new approach, the propagating-source method, is introduced to solve the time-dependent Boltzmann equation. The method relies on the decomposition of the particle distribution function into scattered and unscattered particles. It is assumed in this paper that the particles are transported in a constant-velocity spherically expanding supersonic flow (such as the solar wind) in the presence of a radial magnetic field. Attention too has been restricted to very fast particles. The present paper addresses only large-angle scattering, which is modelled as a BGK relaxation time operator. A subsequent paper (Part 2) will apply the propagating-source method to a small-angle quasilinear scattering operator. Initially, we consider the simplest form of the BGK Boltzmann equation, which omits both adiabatic deceleration and focusing, to re-derive the well-known telegrapher equation for particle transport. However, the derivation based on the propagating-source method yields an inhomogeneous form of the telegrapher equation; a form for which the well-known problem of coherent pulse solutions is absent. Furthermore, the inhomogeneous telegrapher equation is valid for times t much smaller than the ‘scattering time’ τ, i.e. for times t [Lt ] τ, as well as for t > τ. More complicated forms of the BGK Boltzmann equation that now include focusing and adiabatic deceleration are solved. The basic results to emerge from this new approach to solving the BGK Boltzmann equation are the following. (i) Low-order polynomial expansions can be used to investigate particle propagation and transport at arbitrarily small times in a scattering medium. (ii) The theory of characteristics for linear hyperbolic equations illuminates the role of causality in the expanded integro-differential Fokker–Planck equation. (iii) The propagating-source approach is not restricted to isotropic initial data, but instead arbitrarily anisotropic initial data can be investigated. Examples using different ring-beam distributions are presented. (iv) Finally, the numerical scheme can include both small-angle and large-angle particle scattering operators (Part 2). A detailed discussion of the results for the various Boltzmann-equation models is given. In general, it is found that particle beams that experience scattering by, for example, interplanetary fluctuations are likely to remain highly anisotropic for many scattering times. This makes the use of the diffusion approximation for charged-particle transport particularly dangerous under many reasonable solar-wind conditions, especially in the inner heliosphere.