A-SLEIPNNIR: a multiscale, anisotropic adaptive, particle level set framework for moving interfaces. Transport equation applications. We introduce in this paper `A-SLEIPNNIR’ (Adaptive Semi-Lagrangian Ensemble Implementation of Particle level set for Newtonian and non-Newtonian Interfacial Rheology). The proposed method combines a semi-Lagrangian approach for the transport operator, a particle level set technique for interface capturing featuring second-order accurate redistancing, and anisotropic mesh refinement for spatial resolution. The high-order, quadratic finite element discretization, together with an a posteriori error analysis produce a metric tensor that is passed on to an anisotropic mesh generator. We explore the peculiar features of the numerical scheme such as the impact of the marker particles, the reinitialization procedure or the adaptive mesh refinement (AMR) strategy, in a series of benchmark problems for the solution of the interface transport equation with analytically supplied, divergence-free velocity fields. The results show that our approach yields an accurate and robust, yet computationally inexpensive framework for moving interfaces suitable for scientific and engineering applications.