The booming renaissance and rapid development of ultra-dense low earth orbit (LEO) satellite networks (UD-LSNs) are envisioned to realize a giant leap forward for the future sixth generation (6G) coverage expansion, bridging digital divide for remote areas and providing continuous services for user terminals worldwide. However, the inherent dual mobility, massive access scenarios and highly overlapped coverage may trigger frequent, vast and ping-pong handovers, especially with the existing limited and fixed deployment of terrestrial mobility functional entity. To this end, by exploiting the unique opportunity of UD-LSNs, we devise a medium Earth orbit (MEO) assisted distributed hierarchical mobility management architecture (HDMMA) with flexible function configuration to adapt the high dynamic and large scale network. Subsequently, the lightweight handover procedures (LHPs) are proposed for two scenarios under the HDMMA to ensure service continuity, that is on-orbit handover and off-orbit handover. Considering the user mobility attributes and satellite available resources, the on-orbit handover introduces user aggregate to share signaling overhead, while the off-orbit handover is further classified into intra-cluster, inter-cluster and inter-group handover based on the clustering and grouping. Furthermore, we conduct theoretical analysis model on the proposed LHP in terms of signaling overhead and handover latency. Simulation results verify the handover characteristics in UD-LSNs, illustrate the superiority of our HDMMA and demonstrate the handover performance improvement of the proposed LHP.