Recently, there has been a growing interest for mixed categorical meta-models based on Gaussian process (GP) surrogates. In this setting, several existing approaches use different strategies. Among the recently developed methods, we could cite: GP models built using continuous relaxation of the variables, Gower distance based models or GP models derived from direct estimation of the correlation matrix. In this paper, we present a kernel-based approach that extends continuous Gaussian kernels to handle mixed-categorical variables. The proposed kernel leads to a GP surrogate that generalizes continuous relaxation and Gower distance based GP models. The good potential of the proposed framework is shown on analytical mixed-categorical variables test cases. On different settings, our proposed GP models is as accurate as the state-of-the-art GP models.