We use VLT and JWST spectroscopy to investigate the prevalence of galactic outflows in quenching galaxies at cosmic noon. We perform a stacking analysis using Mg II (λ2800 Å) absorption profiles to investigate outflow properties as a function of time since the last major burst of star formation (tburst). We find evidence for high-velocity outflows in the star-forming progenitor population (vout ∼ 1500 ± 450 kms−1), and high velocity (~1000 km/s) outflows seem to be present up to ~1 Gyr after quenching.. The oldest galaxies (tburst > 1 Gyr) show no evidence for significant outflows. Our sample show no signs of AGN in optical observations, suggesting that any AGN in these galaxies have very short duty cycles, and were 'off' when observed. The presence of significant outflows in the older quenched galaxies (tburst > 0.6 Gyr) is difficult to explain with starburst activity, however, and may indicate energy input from episodic AGN activity as the starburst fades. Preliminary analysis of 7 individual recently quenched galaxy spectra at z > 3 taken with JWST NIRSpec reveals 3 out of 7 hosting high velocity outflows in both Mg II and NaD. Only one object is detected in Chandra X-ray observations, further suggesting that AGN duty cycles are short lived at these epochs.