Sun-like stars are well represented in the solar neighbourhood but are currently under-utilised, with many studies of chemical and kinematic evolution focusing on red giants (which can be observed further away) or turn-off stars (which have well measured ages). Recent surveys (e.g. GALAH) provide spectra for large numbers of nearby Sun-like stars, which provides an opportunity to apply our newly developed method for measuring metallicities, temperatures, and surface gravities - the epic algorithm - which yields improved ages via isochrone fitting. We test this on moving groups, by applying it to the large GALAH DR3 sample. This defines a sample of 72288 solar analogue targets for which the stellar parameter measurements are most precise and reliable. These stars are used to estimate, and test the accuracy and precision of, age measurements derived with the SAMD isochrone fitting algorithm. Using these ages, we recover the age-metallicity relationships for nearby (<=1kpc) moving groups, traced by solar analogues, and analyse them with respect to the stellar kinematics. In particular, we found that the age-metallicity relationships of all moving groups follows a particular trend of young (age<6Gyr) stars having constant metallicity and older (age>=6Gyr) stars decreasing in metallicity with increasing age. The Hercules stream carries the highest fraction of metal-rich young stars (~0.1dex) in our sample, which is consistent with a migrating population of stars from the inner Galaxy, and we discuss the possible causes of this migration in the context of our results.