Non-diffracting beams, such as Bessel and Mathieu beams, offer a wide range of potential applications in the fields of bio-photonics, micromanipulation and spectroscopy. One of the main features of these beams is their self-healing behavior where the beams reconstruct after an obstacle. Higher order versions of these beams incorporate non-diffracting optical singularities or vortices propagating together with the beams in a straight line. Vortices are ubiquitous in many parts of physics and their dynamics, especially their creation and annihilation processes are very important in fundamental physics. Newly demonstrated Airy beams represent a different class of non-diffracting beams that do not propagate in a straight line but exhibit a constant transversal acceleration. The self-healing properties of these Airy beams together with their transversal acceleration can be used to optically clear entire regions of microparticles. These Airy beams are created using a spatial light modulator that encodes a cubic phase front on an incident Gaussian beam. Using the same method and suitable computer generated holograms we are able to generate Airy like beams that include optical vortices. In this paper, we study the creation and evolution of Airy beam accelerating vortices from the theoretical and experimental perspective.