Time-domain random-walk (TDRW) algorithms are efficient methods for simulating solute transport along one-dimensional pathways. New extensions of the TDRW algorithm accommodate decay and ingrowth of radionuclides in a decay chain and time-dependent transport velocities. Tests using equilibrium sorption and matrix diffusion retention models demonstrate that the extended TDRW algorithm is accurate and computationally efficient. When combined with stochastic simulation of transport properties, the resulting algorithm, Particle on Random Streamline Segment (PORSS), also captures the effects of random spatial variations in transport velocities, including the effects of very broad velocity distributions. When used in combination with discrete fracture network simulations, the PORSS algorithm provides an accurate and practical method for simulating radionuclide transport at the geosphere scale without invoking the advection-dispersion equation.