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We show that the reformulation of the geometric Robinson-Schensted-Knuth(gRSK) correspondence via local moves, introduced in \cite{OSZ14} can beextended to cases where the input matrix is replaced by more general polygonal,Young-diagram-like, arrays of the form $\polygon$. We also show that arearrangement of the sequence of the local moves gives rise to a geometricversion of the polynuclear growth model (PNG). These reformulations are used toobtain integral formulae for the Laplace transform of the joint distribution ofthe point-to-point partition functions of the log-gamma polymer at differentspace-time points. In the case of two points at equal time $N$ and space atdistance of order $N^{2/3}$, we show formally that the joint law of thepartition functions, scaled by $N^{1/3}$, converges to the two-point functionof the Airy process

Variants of geometric RSK, geometric PNG and the multipoint distribution of the log-gamma polymer