Evolving boundaries are an intrinsic part of many physical processes and numerical methods. Most efforts to model evolving boundaries rely on implicit schemes, such as the level-set method (LSM). LSM provides the means to efficiently model the evolution of a boundary, but lacks the ability to transmit information or provide information directly at the boundary. Explicit alternatives based on remeshing or partial-remeshing are often computationally expensive and inherently complex to implement. This work proposes a solution to this dichotomy: a novel finite element method (FEM) based formulation capable of explicitly discretising moving boundaries in an accurate and numerically-efficient way. It couples the floating node method (FNM) with LSM for the first time, which yield a methodology suitable for implementation as user-element in a generic FEM package. The explicitly discretised boundary allows for a new velocity-extension methodology, and a new LSM-reinitialisation procedure, which show benefits in accuracy and efficiency. The potential of this formulation is showcased within topology optimisation, showing greater geometrical accuracy and improvements in the optimum solution attained when compared to implicit methods.