We report distinct growth regimes of hollow silica fibers formed by hydrodynamic injection of cupric sulfate into silicate solution. The tubes grow either steadily along a continuous jet of buoyant solution or through relaxation oscillations that are governed by chemo-mechanical processes. The dependence of the oscillation period on flow rate and copper concentration is explained in the framework of a simple model. Tailored flow patterns allow the directional control of the tubes and their use as miniature connectors. Our findings are applicable to the understanding of chemical gardens, promise a wealth of nonlinear phenomena, and offer possible applications in microfluidics.