In this talk, we present several generalizations of elastic Brownian motion, analyzing three distinct scenarios that extend the process's classical dynamics. First, we consider a model in which the killing rate is governed by an independent continuous-time Markov chain (CTMC), thereby introducing a switching mechanism for the process’s extinction. Next, we introduce non-exponential delays at the boundary; this extension leads to the appearance of non-local operators in time (for instance, fractional derivatives and convolution-type operators). Finally, we study the case in which the process, instead of being killed, restarts inside the domain via jumps (stochastic restart). The latter dynamics are described by non-local spatial operators, as boundary conditions, related to the jump distribution. For each case, we discuss the associated PDEs and the connection with non-local operators, describing the stochastic dynamics and potential physical applications.