Dynamic deep neural networks, particularly multi-exit networks, are increasingly recognized for their efficiency in edge-cloud scenarios. However, they are vulnerable to latency attacks that can degrade performance by increasing computation time. Current attack strategies often require white-box access to the model or lead to significant drops in inference accuracy, making them easily detectable. This paper introduces SPLAT, a novel approach for executing stealthy and practical latency attacks on dynamic multi-exit models under black-box conditions. SPLAT employs a two-stage mechanism: the first stage generates coarse-grained attack inputs using a functional surrogate model, while the second stage refines these perturbations through an efficient query strategy to enhance stealthiness and effectiveness. Extensive experiments validate that SPLAT significantly outperforms existing methods across various models and datasets.