We consider the state of a one-dimensional critical quantum system after a \thermal cut-and-glue quench", which is a local quench in which two independently thermalized halves are connected to form a homogeneous in nite system and left to evolve unitarily until they reach a non-equilibrium steady state (NESS). This quench was studied in [1], and exact CFT results for the current and its uctuations in the NESS were found. We add to these results by studying the growth of entanglement after the quench. Furthermore, we generalise to the case in which the system is not critical, but described by an integrable relativistic quantum eld theory (IQFT) with diagonal scattering, and nd exact expressions for the energy current and scaled cumulant generating function (CGF) in the NESS. Another generalisation we consider is the thermal cut-and-glue quench for N independently thermalised critical one-dimensional systems. These are made to form a quantum junction consisting of N one-dimensional critical systems that are connected at one point in a star graph con guration, with a particular, simple connection condition at the vertex. We nd in this situation the exact energy current and scaled CGF in the NESS. This thesis is based on the publications [2{4], which are incorporated in their published form in Chapters 3 and 4.