This work investigates the real-time performance of Linux kernels and distributions with a PREEMPT_RT real-time patch on ARM-based embedded devices. Experimental measurements, which are mainly based on heuristic methods, provide novel insights into Linux real-time performance on ARM-based embedded devices (e.g., BeagleBoard and RaspberryPi). Evaluations of the Linux real-time performance are based on specific real-time software measurement modules, developed for this purpose, and the use of a standard benchmark tool, cyclictest. Software modules were designed upon the introduction of a new response task model, an innovative aspect of this work. Measurements include the latency of response tasks at user and kernel space, the response on the execution of periodic tasks, the maximum sustained frequency and general latency performance metrics. The results show that in such systems the PREEMPT_RT patch provides more improved real-time performance than the default Linux kernels. The latencies and particularly the worst-case latencies are reduced with real-time support, thus making such devices running Linux with PREEMPT_RT more appropriate for use in time-sensitive embedded control systems and applications. Furthermore, the proposed performance measurements approach and evaluation methodology could be applied and deployed on other Linux-based real-time platforms.