Which nodes will earn fees for verifying a particular transaction? When will a node stop earning fees for verifying transactions?

In the Tempo Ledger each transaction is officially verified by a 'temporal proof path' which consists of a chain of nodes. 

The nodes which are taking part in the temporal proof, that have verifiably done work for the network and as such are rewarded with income through transaction fees and node earnings at the end of the earnings cycle.Which nodes take part in the temporal proof is random. Therefore, each node is expected to gain income roughly proportional to the activity in the network and the fraction of the network that they are supporting.

For more information on Temporal Proofs, see: https://papers.radixdlt.com/tempo/#temporal-proof-provisioning

For more information on Incentives, see: https://papers.radixdlt.com/incentives/#incentives