Block-coordinate algorithms are recognized to furnish efficient iterative schemes for addressing large-scale problems, especially when the computation of full derivatives entails substantial memory requirements and computational efforts. In this paper, we propose a randomized block proximal gradient algorithm for minimizing the sum of a smooth function and a separable proper lower semicontinuous function, both possibly nonconvex. In contrast to previous work, we only assume that the partial gradients of the smooth function are locally Lipschitz continuous with respect to their block of coordinates. At each iteration, the method adaptively selects a proximal stepsize to satisfy a sufficient decrease condition without requiring knowledge of the local Lipschitz moduli of the partial gradients of the smooth function. Hence, our work extends the applicability of randomized block-coordinate descent methods to more general problems where global Lipschitz gradient assumptions are not satisfied. In addition, our algorithm incorporates the possibility of conducting an additional boosted linesearch to enhance the performance of the algorithm. Our main result establishes subsequential convergence to a stationary point of the problem almost surely. Numerical experiments in nonnegative matrix factorization report that our method competes well with recognized algorithms. Furthermore, a symmetric variant of nonnegative matrix factorization illustrates the advantage of the boosted linesearch in problems where the proximal stepsize cannot be tightly estimated in accordance to the local Lipschitz modulus.