  We extend the techniques introduced in \cite{DoMaB1} for contractible Riemann
surfaces to construct minimal Lagrangian immersions from arbitrary Riemann
surfaces into $\mathbb{C}P^2$ via the loop group method. Based on the
potentials of translationally equivariant minimal Lagrangian surfaces, we
introduce perturbed equivariant minimal Lagrangian surfaces in $\mathbb{C}P^2$
and construct a class of minimal Lagrangian cylinders. Furthermore, we show
that these minimal Lagrangian cylinders approximate Delaunay cylinders with
respect to some weighted Wiener norm of the twisted loop group $\Lambda
SU(3)_{\sigma}$.
