Quantification of electronic and magnetoelastic mechanisms of first-order magnetic phase transitions from first principles: application to caloric effects in La(Fe$_x$Si$_{1-x}$)$_{13}$

La(Fe$_x$Si$_{1-x}$)$_{13}$ and derived quaternary compounds are well-known for their giant, tunable, magneto- and barocaloric responses around a first-order paramagnetic-ferromagnetic transition near room temperature with low hysteresis. Remarkably, such a transition shows a large spontaneous volume change together with itinerant electron metamagnetic features. While magnetovolume effects are well-established mechanisms driving first-order transitions, purely electronic sources have a long, subtle history and remain poorly understood. Here we apply a disordered local moment picture to quantify electronic and magnetoelastic effects at finite temperature in La(Fe$_x$Si$_{1-x}$)$_{13}$ from first-principles. We obtain results in very good agreement with experiment and demonstrate that the magnetoelastic coupling, rather than purely electronic mechanisms, drives the first-order character and causes at the same time a huge electronic entropy contribution to the caloric response.