Some alterna-coins may not rely on the same sort of processing power as Bitcoin. For example, Bitcoin mining has migrated to specialized GPUs and ASICs. Litecoin specifically chose 'scrypt' memory-hard hashing, for which the same GPUs/ASICs provide no benefit. Other 'proof-of-stake' proposals for maintaining the ledger emphasize staying actively connected to the network – pulling your own weight in verification and distribution – rather than raw hashing power.
Bitcoin's proof-of-work-during-verification trick is neat, but may not be the only way, and thus viable alternatives may emerge that don't face the barrier of an early miner decision to redeploy computational power elsewhere.