Contractile actomyosin network flows are crucial for many cellular processes including cell division and motility, morphogenesis and transport. How local remodeling of actin architecture tunes stress production and dissipation and regulates large-scale network flows remains poorly understood. Here, we generate contracting actomyosin networks with rapid turnover in vitro, by encapsulating cytoplasmic Xenopus egg extracts into cell-sized 'water-in-oil' droplets. Within minutes, the networks reach a dynamic steady-state with continuous inward flow. The networks exhibit homogeneous, density-independent contraction for a wide range of physiological conditions, implying that the myosin-generated stress driving contraction and the effective network viscosity have similar density dependence. We further find that the contraction rate is roughly proportional to the network turnover rate, but this relation breaks down in the presence of excessive crosslinking or branching. Our findings suggest that cells use diverse biochemical mechanisms to generate robust, yet tunable, actin flows by regulating two parameters: turnover rate and network geometry.