Chemical gardens are macroscopic structures that form when a salt seed is submerged in an alkaline solution. Their thin precipitate membranes separate the reactant partners and slow down the approach toward equilibrium. During this stage, a gradual thickening occurs, which is driven by steep cross-membrane gradients and governed by selective ion transport. We study these growth dynamics in microfluidic channels for the case of Ni(OH)2 membranes. Fast flowing reactant solutions create thickening membranes of a nearly constant width along the channel, whereas slow flows produce wedge-shaped structures that fail to grow along their downstream end. The overall dynamics and shapes are caused by the competition of reactant consumption and transport replenishment. They are reproduced quantitatively by a two-variable reaction-diffusion-advection model which provides kinetic insights into the growth of precipitate membranes.