We initiate the study of how to extend the correspondence between dimer models and (0+1)-dimensional cluster integrable systems to (1+1) and (2+1)-dimensional continuous integrable field theories, addressing various points that are necessary for achieving this goal. We first study how to glue and split two integrable systems, from the perspectives of the spectral curve, the resolution of the associated toric Calabi-Yau 3-folds and Higgsing in quiver theories on D3-brane probes. We identify a continuous parameter controlling the decoupling between the components and present two complementary methods for determining the dependence on this parameter of the dynamical variables of the integrable system. Interested in constructing systems with an infinite number of degrees of freedom, we study the combinatorics of integrable systems built up from a large number of elementary components, and introduce a toy model capturing important features expected to be present in a continuous reformulation of cluster integrable systems. Show Partial Abstract