We report temperature dependent measurements of ambient pressure specific heat, magnetic susceptibility, anisotropic resistivity and thermal expansion as well as in-plane resistivity under pressure up to 20.8 kbar on single crystals of EuAg$_4$As$_2$. Based on thermal expansion and in-plane electrical transport measurements at ambient pressure this compound has two, first order, structural transitions in 80 - 120 K temperature range. Ambient pressure specific heat, magnetization and thermal expansion measurements show a cascade of up to seven transitions between 8 and 16 K associated with the ordering of the Eu$^{2+}$ moments. In-plane electrical transport is able to detect more prominent of these transitions: at 15.5, 9.9, and 8.7 K as well as a weak feature at 11.8 K at ambient pressure. Pressure dependent electrical transport data show that the magnetic transitions shift to higher temperatures under pressure, as does the upper structural transition, whereas the lower structural transition is suppressed and ultimately vanishes. A jump in resistivity, associated with the upper structural transition, decreases under pressure with an extrapolated disappearance (or a change of sign) by 30-35 kbar. In the 10 - 15 kbar range a kink in the pressure dependence of the upper structural transition temperature as well as the high and low temperature in-plane resistivities suggest that a change in the electronic structure may occur in this pressure range. The results are compared with the literature data for SrAg$_4$As$_2$.
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