We investigate ongoing accretion activity in young stars in the TW Hydrae association (TWA, ~8-10 Myr), an ideal target to probe the final stages of disk accretion down to brown dwarf masses. Our sample comprises eleven TWA members with infrared excess, amounting to 85% of the total TWA population with disks, with spectral types between M0 and M9, and masses between 0.58 and 0.02 Msol. We employed homogeneous spectroscopic data from 300 to 2500 nm, obtained with X-shooter, to derive individual extinction, stellar parameters, and accretion parameters simultaneously. We then examined Balmer lines and forbidden emission lines to probe the physics of the star-disk interaction environment. We detected signatures of ongoing accretion for 70% of our TWA targets. This implies a fraction of accretors of 13-17% across the entire TWA (accounting for potentially accreting members not included in our survey). The spectral emission associated with these stars reveals a more evolved stage of these accretors compared to younger PMS populations: (i) a large fraction (~50%) exhibit nearly symmetric, narrow H_alpha line profiles; (ii) over 80% exhibit Balmer decrements consistent with moderate accretion and optically thin emission; (iii) less than a third exhibit forbidden line emission in [O I] 6300A, indicative of winds and outflows activity. However, the distribution in accretion rates (Mdot) derived for the TWA sample follows closely that of younger regions for Mstar~0.1-0.3 Msun. An overall correlation between Mdot and Mstar is detected, best reproduced by Mdot~Mstar^(2.1+/-0.5). At least in the lowest Mstar regimes, stars that still retain a disk at ages ~8-10 Myr are found to exhibit statistically similar, albeit moderate, accretion levels as those measured around younger objects. This slow Mdot evolution may be associated with longer evolutionary timescales of disks around low-mass stars.