Foldcore sandwich structures offer a promising alternative to conventional honeycomb sandwiches in the field of lightweight structures, demonstrating significant potential as efficient energy absorption devices. The dynamic behavior of foldcore sandwich structures is crucial in response to low-velocity impacts in various engineering scenarios such as bird strikes, airdrops, and vehicle collisions. This study investigates the dynamic responses of a kirigami-inspired pyramid foldcore sandwich subjected to low-velocity impacts, which has previously exhibited remarkable energy absorption efficiency under quasi-static compression. Through a combination of experimental investigations and numerical simulations under impact conditions, it is observed that the pyramid foldcore initially undergoes pronounced localized deformation which results in a sensitivity to the loading rate and causes the high initial peak stress. Different from the mechanism observed under quasi-static compression, additional stationary plastic hinges on the facets are triggered during the post-buckling stage, thereby slightly enhancing the overall energy absorption. Moreover, the multi-layer pyramid foldcores with graded geometries are proposed, characterized by varying height and wall thickness for each layer. The graded pyramid foldcores significantly reduce the initial peak stress while maintaining energy absorption efficiency. In comparison with the conventional square honeycomb and Miura-ori foldcore under the impact velocity of 10 m/s, the uniformity ratio of the graded pyramid foldcore decreases by 70.9 % and 80.5 %, while the specific energy absorption improves by 0.97 % and 138.37 %, respectively. To summarize, the graded pyramid foldcore shows outstanding energy absorption efficiency, indicating its potential as a high-performance sandwich structure for impact applications.