In this paper, a novel hardware architecture for neuroevolution is presented, aiming to enable the continuous adaptation of systems working in dynamic environments, by including the training stage intrinsically in the computing edge. It is based on the block-based neural network model, integrated with an evolutionary algorithm that optimizes the weights and the topology of the network simultaneously. Differently to the state-of-the-art, the proposed implementation makes use of advanced dynamic and partial reconfiguration features to reconfigure the network during evolution and, if required, to adapt its size dynamically. This way, the number of logic resources occupied by the network can be adapted by the evolutionary algorithm to the complexity of the problem, the expected quality of the results, or other performance indicators. The proposed architecture, implemented in a Xilinx Zynq-7020 System-on-a-Chip (SoC) FPGA device, reduces the usage of DSPs and BRAMS while introducing a novel synchronization scheme that controls the latency of the circuit. The proposed neuroevolvable architecture has been integrated with the OpenAI toolkit to show how it can efficiently be applied to control problems, with a variable complexity and dynamic behavior. The versatility of the solution is assessed by also targeting classification problems.