The evolution of influenza viruses is fundamentally shaped by within-host processes. However, the within-host evolutionary dynamics of influenza viruses remain incompletely understood, in part because most studies have focused on infections in healthy adults based on single timepoint data. Here, we analysed the within-host evolution of 82 longitudinally-sampled individuals, mostly young children, infected with A/H1N1pdm09 or A/H3N2 viruses between 2007 and 2009. For A/H1N1pdm09 infections during the 2009 pandemic, nonsynonymous minority variants were more prevalent than synonymous ones. For A/H3N2 viruses in young children, early infection was dominated by purifying selection. As these infections progressed, nonsynonymous variants typically increased in frequency even when within-host virus titres decreased. Unlike the short-lived infections of adults where de novo within-host variants are rare, longer infections in young children allow for the maintenance of virus diversity via mutation-selection balance creating potentially important opportunities for within-host virus evolution.