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HLR vs VLR
The HLR is the permanent home record; the VLR is the temporary copy held by whichever MSC currently serves the subscriber. For a mobile-terminated SMS, the delivering SMSC first asks the HLR where the subscriber is, then sends MT-ForwardSM to that serving MSC — whose VLR knows how to reach the handset. See /answers/what-is-hlr and /answers/what-is-msc.
Where the VLR sits in SMS delivery
SendRoutingInfoForSM to the HLR returns the address of the serving MSC; the SMSC then forwards the message to that MSC, and its VLR resolves the current radio context to page the handset. The VLR is invisible to the sender but central to why cross-operator delivery works at all. See /answers/ss7-map-send-routing-info-for-sm.