According to the Findlaw Blotter, criminals who flee a state to avoid punishment have a warrant issued for their arrest, and the statute of limitations for the crime is not affected by fleeing. The fleeing criminal's reputation also may be damaged due to their mug shot being shared across social media by various law enforcement agencies.
The Findlaw Blotter points out a number of other possible effects for criminals and alleged criminals who flee a state to avoid punishment. One such possibility is the use of bounty hunters to track the criminal down. Bounty hunters are legally authorized to hunt fleeing criminals and also have a financial incentive to catch them, so fleeing criminals should be aware that this increases the number of interested parties who are hunting them down.
Additionally, depending on the state to which the criminal has fled, they may face extradition back to the jurisdiction of the original crime. In order to extradite, the state must provide evidence that the criminal deliberately fled in order to avoid punishment. Anyone considering fleeing a state to escape punishment should have a conversation with a lawyer so they are fully versed in how factors such as extradition would affect their situation.