When accelerating an index cluster migration to new hardware using CLI commands to adjust 'max_peer_build_load' and 'max_peer_rep_load', key considerations include network latency and storage Input/Output Operations Per Second (IOPS). These factors directly impact the speed and efficiency of data transfer between nodes. High network latency can slow down communication, while sufficient storage IOPS ensures the storage system can handle the increased load. Hence, optimizing both parameters is crucial for a smooth and fast migration process.