In the usual case, both the seizure and the search must be reasonable under the Fourth Amendment in order for the evidence to be admissible. The U.S. Supreme Court and federal appeals courts have considered both issues when officers have used K-9s to detect contraband.
So should you chase or contain? It depends on the situation and what agency you work for. Just as there are a variety of threats to consider about suspects in flight, so too are there differences of opinion as to how to address the problem.
As the role of the police dog expands, training has taken on more importance, even as economic times threaten the funding for the dogs and their handlers.