Edge devices operate in constrained and varying resource settings, requiring dynamic architectures that can adapt to limitations of the available resources. To meet such demands, layer dropping ($\mathcal{LD}$) approach is typically used to transform static models into dynamic ones by skipping parts of the network along with reducing overall computational complexity. However, existing $\mathcal{LD}$ methods greatly impact the dynamic model's performance for low and high dropping cases, deteriorating the performance-computation trade-off. To this end, we propose a distillation-based layer dropping (DLD) framework that effectively combines the capabilities of knowledge distillation and $\mathcal{LD}$ in an end-to-end fashion, thereby achieving state-of-the-art performance for dynamic speech networks. Comprehensive experimentation utilizing well-known speech recognition methods, including conformer and WavLM, on three public benchmarks demonstrates the effectiveness of our framework, reducing the word error rate by $9.32\%$ and $2.25\%$ for high and no dropping cases with $33.3\%$ reduction in training time.