The overall response of elasto-viscoplastic polycrystals demonstrates non-Maxwellian, long-memory effects, even when the constituent single crystals exhibit Maxwellian, short-memory behaviour. To characterize this emergent behaviour, this work presents analytical homogenization estimates for the elasto-viscoplastic polycrystals and demonstrates that large viscoplastic nonlinearity or anisotropy for the single crystals lead to strong long-term memory effects on the overall response, which cannot be captured with a Maxwellian model for the overall response. We derive differential equations for the time evolution of the stress averages and intragranular fluctuations, which in turn yield the macroscopic response, by means of the differential variational principle (DVP) [1] that combines the incremental variational procedure [2] to deal with long-term memory effects and `second-order' linearization schemes to handle viscous nonlinearity. Comparisons with the Maxwellian approximation for the overall response, obtained by separately homogenizing the elastic and viscoplastic components, serve to characterize the long memory effect. It is found that the strong long memory effects for polycrystals consisting of nonlinear or anisotropic grains are caused by rapid changes in the inter- and intra-granular field fluctuations. While for face-centered cubic polycrystals, the long memory effect is rather minimal under creep loading, for ice-like hexagonal close-packed (HCP) polycrystals the effect is quite significant. For constant overall strain-rate loading, significant improvements over the Maxwellian approximation are also observed for the HCP polycrystals, especially in the transient regime. To assess their accuracy, the new estimates are compared with available full-field and experimental results from the literature and good agreements are generally found even for polycrystals with highly nonlinear or anisotropic grains. Overall, the DVP is able to accurately characterize the transient response of low-symmetry polycrystals exhibiting strong non-Maxwellian behavior. References [1] Das, S., Ponte Castaneda, P.. Journal of the Mechanics and Physics of Solids, Vol. 147, p. 104202, 2021. [2] Agoras, M and Avazmohammadi, R, Ponte Castaneda, P.. International Journal of Solids and Structures, Vol. 97, p. 668--686, 2016.