  The aim of this paper is to develop a method for proving almost sure
convergence in Gromov-Hausodorff-Prokhorov topology for a class of models of
growing random graphs that generalises R\'emy's algorithm for binary trees. We
describe the obtained limits using some iterative gluing construction that
generalises the famous line-breaking construction of Aldous' Brownian tree.
  In order to do that, we develop a framework in which a metric space is
constructed by gluing smaller metric spaces, called \emph{blocks}, along the
structure of a (possibly infinite) discrete tree. Our growing random graphs
seen as metric spaces can be understood in this framework, that is, as evolving
blocks glued along a growing discrete tree structure. Their scaling limit
convergence can then be obtained by separately proving the almost sure
convergence of every block and verifying some relative compactness property for
the whole structure. For the particular models that we study, the discrete tree
structure behind the construction has the distribution of an affine
preferential attachment tree or a weighted recursive tree. We strongly rely on
results concerning those two models of random trees and their connection,
obtained in a companion paper.
