For many environmental processes, recent studies have shown that the dependence strength is decreasing when quantile levels increase. This implies that the popular max-stable models are inadequate to capture the rate of joint tail decay, and to estimate joint extremal probabilities beyond observed levels. We here develop a more flexible modeling framework based on the class of max-infinitely divisible processes, which extend max-stable processes while retaining dependence properties that are natural for maxima. We propose two parametric constructions for max-infinitely divisible models, which relax the max-stability property but remain close to some popular max-stable models obtained as special cases. The first model considers maxima over a finite, random number of independent observations, while the second model generalizes the spectral representation of max-stable processes. Inference is performed using a pairwise likelihood. We illustrate the benefits of our new modeling framework on Dutch wind gust maxima calculated over different time units. Results strongly suggest that our proposed models outperform other natural models, such as the Student-t copula process and its max-stable limit, even for large block sizes.