X-ray coherent diffractive imaging (CDI) techniques have been applied with widespread impact to study nanoscale material properties. New fast framing detectors may reveal dynamics that occur at millisecond timescales. This work demonstrates by simulation that kilohertz synchrotron CDI is possible, by making use of redundant information from static parts of the image field. Reconstruction ambiguities are strongly suppressed by applying a spatiotemporal constraint, obviating the need for slower methods of introducing diversity such as ptychography. The relationship between image fidelity and time resolution is investigated and shows that dynamics an order of magnitude faster can be reconstructed, compared with conventional CDI.