The application of the Sliding Frank-Wolfe algorithm to gridless compressive beamforming is investigated for single and multi-snapshot measurements and the estimation of the three-dimensional (3D) position of the sources and their amplitudes. Sources are recovered by solving an infinite dimensional optimization problem, promoting sparsity of the solutions, and avoiding the basis mismatch issue. The algorithm does not impose constraints on the source model or array geometry. A variant of the algorithm is proposed for greedy identification of the sources. The experimental results and Monte Carlo simulations in 3D settings demonstrate the performances of the method and its numerical efficiency compared to the state of the art.