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We have studied the dynamics in the sunspot transition region between the chromosphere and the corona and investigated the extension of the flow field into the corona. Based on EUV spectra of a medium size sunspot and its surroundings, NOAA 7981, observed with CDS and SUMER on SOHO, we derive line-of-sight velocities and study the line profiles for a series of emission lines.The flow field in the low corona is found to differ markedly from that in the transition region. In the transition region the relative line-of-sight velocity shows an upflow in the umbra and relatively large areas with downflow that cover part of the penumbra. The spatial extent of these areas with upflow and downflow increases with increasing temperature in the transition region, but the whole flow field changes character as the temperature increases from the upper transition region to the low corona. Based on a calibration of the SUMER wavelength scale we find that the entire sunspot transition zone appears to be moving downwards towards the chromosphere. The relation between this finding and the general tendency for transition-region lines to show a net red shift is discussed.Several of the transition-region spectral line profiles are observed to show two line components with Gaussian shape and line-of-sight velocities that differ markedly. Several of the line profiles that are composed of two spectral line components occur close to the dividing line between up- and downflow. A discussion of this observation is presented. In small regions with spatial extent of a few arc sec we detect enhanced continuum emission underlying explosive events. The similarities between explosive events with continuum emission and the moustaches observed in Hα close to sunspots are so striking that we are tempted to introduce the notation ‘transition-region moustaches’.