Image guidance in radiotherapy and extracranial radiosurgery offers the potential for precise radiation dose delivery to a moving tumour. Recent work has demonstrated how to locate and track the position of a tumour in real-time using diagnostic x-ray imaging to find implanted radio-opaque markers. However, the delivery of a treatment plan through gating or beam tracking requires adequate consideration of treatment system latencies, including image acquisition, image processing, communication delays, control system processing, inductance within the motor, mechanical damping, etc. Furthermore, the imaging dose given over long radiosurgery procedures or multiple radiotherapy fractions may not be insignificant, which means that we must reduce the sampling rate of the imaging system. This study evaluates various predictive models for reducing tumour localization errors when a real-time tumour-tracking system targets a moving tumour at a slow imaging rate and with large system latencies. We consider 14 lung tumour cases where the peak-to-peak motion is greater than 8 mm, and compare the localization error using linear prediction, neural network prediction and Kalman filtering, against a system which uses no prediction. To evaluate prediction accuracy for use in beam tracking, we compute the root mean squared error between predicted and actual 3D motion. We found that by using prediction, root mean squared error is improved for all latencies and all imaging rates evaluated. To evaluate prediction accuracy for use in gated treatment, we present a new metric that compares a gating control signal based on predicted motion against the best possible gating control signal. We found that using prediction improves gated treatment accuracy for systems that have latencies of 200 ms or greater, and for systems that have imaging rates of 10 Hz or slower.