We present a new set of models for intermediate-mass asymptotic giant branch (AGB) stars (4.0, 5.0, and 6.0 M-circle dot) at different metallicities (-2.15 <= [Fe/H] <= +0.15). This set integrates the existing models for low-mass AGB stars (1.3 <= M/M-circle dot <= 3.0) already included in the FRUITY database. We describe the physical and chemical evolution of the computed models from the main sequence up to the end of the AGB phase. Due to less efficient third dredge up episodes, models with large core masses show modest surface enhancements. This effect is due to the fact that the interpulse phases are short and, therefore, thermal pulses (TPs) are weak. Moreover, the high temperature at the base of the convective envelope prevents it from deeply penetrating the underlying radiative layers. Depending on the initial stellar mass, the heavy element nucleosynthesis is dominated by different neutron sources. In particular, the s-process distributions of the more massive models are dominated by the Ne-22(alpha, n)Mg-25 reaction, which is efficiently activated during TPs. At low metallicities, our models undergo hot bottom burning and hot third dredge up. We compare our theoretical final core masses to available white dwarf observations. Moreover, we quantify the influence intermediate-mass models have on the carbon star luminosity function. Finally, we present the upgrade of the FRUITY web interface, which now also includes the physical quantities of the TP-AGB phase for all of the models included in the database (ph-FRUITY).