In recent years, the heterogeneity of both commodity and supercomputers hardware has increased sharply. Accelerators, such as GPUs or Intel Xeon Phi co-processors, are often key to improving speed and energy efficiency of highly-parallel codes. However, due to the complexity of heterogeneous architectures, optimization of codes for a certain type of architecture as well as porting codes across different architectures, while maintaining a comparable level of performance, can be extremely challenging. Addressing the challenges associated with performance optimization and performance portability, autotuning has gained a lot of interest. Autotuning of performance-relevant source-code parameters allows to automatically tune applications without hard coding optimizations and thus helps with keeping the performance portable. In this paper, we introduce a benchmark set of ten autotunable kernels for important computational problems implemented in OpenCL or CUDA. Using our Kernel Tuning Toolkit, we show that with autotuning most of the kernels reach near-peak performance on various GPUs and outperform baseline implementations on CPUs and Xeon Phis. Our evaluation also demonstrates that autotuning is key to performance portability. In addition to offline tuning, we also introduce dynamic autotuning of code optimization parameters during application runtime. With dynamic tuning, the Kernel Tuning Toolkit enables applications to re-tune performance-critical kernels at runtime whenever needed, for example, when input data changes. Although it is generally believed that autotuning spaces tend to be too large to be searched during application runtime, we show that it is not necessarily the case when tuning spaces are designed rationally. Many of our kernels reach near peak-performance with moderately sized tuning spaces that can be searched at runtime with acceptable overhead. Finally we demonstrate, how dynamic performance tuning can be integrated into a real-world application from cryo-electron microscopy domain.