Foam-filled honeycombs have been widely applied due to their excellent load transfer mitigation and energy absorption capacity. In the present study, a layered graded foam concrete-filled auxetic honeycomb was proposed by tuning its overall compression deformation mode to layer-by-layer deformation mode to realize multi-level structural protection. The effect of the honeycomb cell-wall thickness gradient (with an average thickness of 0.25 mm, thickness gradients of 0.30:0.25:0.20, 0.35:0.25:0.15 and 0.40:0.25:0.10, and corresponding positive gradients) and the foam concrete filler density gradient (408:575:848, 848:575:408) on the response mode, load transfer, energy absorption, and Poisson’s ratio of the proposed composite was systematically investigated. The results showed that the graded composite exhibited an obvious layered deformation mode and a negative Poisson’s ratio effect under relatively low and moderate loading rates (1 m/s, 10 m/s, respectively), especially with the foam concrete density gradient. Under a high loading rate (100 m/s), the graded composite demonstrated progressive collapse initiating from the loading end with a layer-by-layer crushing mode, regardless of the thickness and density gradient. In the response of the composite with a 0.2:0.2:0.2 thickness ratio and a 408:575:848 foam concrete gradient subjected to 1 m/s crushing, the first-layer, second-layer, and third-layer foam concrete absorbed 94.62%, 88.72%, and 86.94% of the total foam concrete energy absorption in the corresponding crushing stage, respectively. Compared with the counterpart homogeneous composites, although the graded composite had an insignificant improvement on energy absorption (less than 5%), it was able to significantly reduce the peak load (as high as 30%) to mitigate the load transfer to the protected structure. The effective Poisson’s ratio of the first layer in the composite with positive gradient (408:575:848) increased to −2 then converged to −0.6 under 2 m/s and 10 m/s crushing, and ranged from −0.4 to −0.1 under 50 m/s and 100 m/s crushing, respectively. The effective Poisson’s ratio of the middle and bottom layers increased to −2 initially and converged to range −0.4 to −0.1, regardless of the crushing speed. The staged response mode of the graded composite facilitated the realization of multi-level structure protection with significantly reduced peak load transferred to the protected structure and tuned energy absorption.