  Rail-yard graphs are a general class of graphs introduced in \cite{bbccr} on
which the random dimer coverings form Schur processes. We study asymptotic
limits of random dimer coverings on rail yard graphs with free boundary
conditions on both the left boundary and the right boundary (double-sided free
boundary) when the mesh sizes of the graphs go to 0. Each dimer covering
corresponds to a sequence of interlacing partitions starting with an arbitrary
partition and ending in an arbitrary partition. Under the assumption that the
probability of each dimer covering is proportional to the product of weights of
present edges, we obtain the moment formula for the height function which
includes an infinite product. By passing down to the scaling limit, we compute
the limit shape (law of large numbers) of the rescaled height functions and
prove the convergence of unrescaled height fluctuations to a diffeomorphic
image of the restriction of the 0-boundary Gaussian free field (central limit
theorem) on the upper half plane to a subset. Applications include the limit
shape and height fluctuations for free boundary steep tilings as proposed in
\cite{BCC17}. The technique to obtain these results is to analyze a class of
Macdonald processes with dual specializations, subject to further complexities
arising from the infinite product in the moment formula.
  We also obtain a new algorithm to sample double-sided free boundary dimer
coverings on rail-yard graphs, which fulfills an open problem in
\cite{bbbccv14}.
