When the packet reaches the Egress router, the MPLS TTL value is copied into the IP TTL field (except when the MPLS TTL value is higher than the existing IP TTL value to prevent loops)
In a label stack, an LSR only modifies the top label and doesn’t modify the other labels below it.
When the TTL reaches 1, the TTL expires and the packet is dropped. In classic IP networks, an ICMP “Time exceeded” message would be sent back to the original sender. In MPLS, it can happen that the router that dropped the packet doesn’t know how to reach the source of the packet. To address this, the router sends the ICMP message towards the destination, in order to find the Egress router. The Egress router should know how to reach the source of the dropped packet and it will now send the ICMP packet on the path to the source. (it may very well go through the router that dropped the packet initially).