Propensity score matching linearity assumption

One of the advantage that i have herd talked about propensity score matching, vs. a regression, is that propensity-score matching doesn't rely on linearity assumptions.

This seems incorrect on the surface, given probit [/logit] models used to calculate the propensity scores, will embed linearity assumptions. [i.e assuming that you must estimate propensity scores]

What would be references to material which discuss this, or compare how similar the assumptions of matching and regressions are more broadly?

Second, propensity scores are most generally the predictions from a model that estimates probabilities conditional on covariates. Nothing prevents you from using non-linear models, such as random forest, boosted regression trees, neural networks, etc. In fact, there is a package for R called twang developed by the RAND Corporation that uses the gbm package to optimize the tuning parameters of boosted regression trees with respect to user-specified sample-balance criteria, then output the predicted propensity scores for your sample using the balance-optimized model. You can also specify whether you are after an average treatment effect (ATE) on the population or on the treated (ATT). I wrote a silly little LinkedIn post that gives some brief pointers on how to use twang-generated propensity scores as the distance metric for nearest-neighbor matching as implemented in the MatchIt package.