One-dimensional critical integrable lattice models with a finite-dimensional degree of freedom attached to each lattice site can exhibit the remarkable phenomenon of the emergence of continuous degrees of freedom in the scaling limit. The most studied system in which this occurrence has been observed is the staggered six-vertex model with twisted boundary conditions. Here, we study the influence of different boundary conditions and higher-rank generalisations on/of this model. Our analysis is based on the Bethe ansatz and the formulation of a conserved quantity — the so-called quasi-momentum — which parametrises the non-compact degree of freedom directly on the lattice. Moreover, we carry over the powerful approach of the correspondence between ordinary differential equations and integrable quantum field theories to the case of open boundary conditions. Using this methodology, we can fully classify the scaling limit of the alternating six-vertex model with Uq(sl(2)) boundary conditions. Furthermore, we demonstrate the incompatibility of antidiagonal boundary conditions with a non-compact degree of freedom in the scaling limit. Finally, we provide numerical evidence that the higher rank generalisation, the D(2)3 model, possesses two independent continuous components.