The challenge of processing heavy-load ML tasks, particularly CNN-based ones at resource-constrained IoT devices, has encouraged the use of edge servers. The edge offers performance levels higher than the end devices and better latency and security levels than the Cloud. On top of that, the rising complexity of ML applications, the ever-increasing number of connected devices, and the current demands for energy efficiency require optimizing such CNN models. Pruning and early-exit are notable optimizations that have been successfully used to alleviate the computational cost of inference. However, these optimizations have not yet been exploited simultaneously: while pruning is usually applied at design time, which involves retraining the CNN before deployment, early-exit is inherently dynamic. In this work, we propose AdaPEx, a framework that exploits the intrinsic reconfigurable FPGA capabilities so both can be cooperatively employed. AdaPEx first explores the trade-off between pruning and early-exit at design-time, creating a design space never exploited in the state-of-the-art. Then, AdaPEx applies FPGA reconfiguration as a means to enable the combined use of pruning and early-exit dynamically. At runtime, this allows matching the inference processing to the current edge conditions and a user-configurable accuracy threshold. In a smart IoT application, AdaPEx processes up to 1.32× more inferences and improves EDP by up to 2.55× over the state-of-the-art FPGA-based FINN accelerator.