Order-of-addition experiments arise when the response depends on the order in which components are added. Existing linear models provide interpretable baselines, but they may fail to capture nonlinear relationships and complex structures, which motivates the use of Gaussian process models. In this paper, we build upon Gaussian process modeling for order-of-addition experiments and develop a design framework that employs Kendall's $\tau$ distance to quantify dissimilarity among component-addition orders. Motivated by the asymptotic D-optimality of maximin Kendall's $\tau$ distance designs, we establish an upper bound for the minimum Kendall's $\tau$ distance as a benchmark, and propose construction methods to generate flexible designs with favorable properties. These space-filling designs exhibit robust performance even when the true model is unknown or misspecified. Numerical studies demonstrate that Kendall's $\tau$ distance-based Gaussian process model and maximin Kendall's $\tau$ distance designs perform effectively across a broad range of scenarios.