IIA assumption: difference logit and probit

Considering the following question about the Independence of Irrelevant Alternatives assumption:

Alternatives to multinomial logistic regression

It seems as if IIA is only a problem when using a multinomial logit model, but, as the answers seems to imply, this assumption is not necessary for nested probit models and/or mixed multinomial logit.

Why is this the case? What about nested logit models? It seems as if the IIA assumption is only a problem for logit but is not necessary for probit models. Is this true?

A violation of the IIA assumption is basically a case of correlation between the residuals for the equations predicting each of the $k-1$ categories (excluding the baseline) of the response variable. A classic example is when you have two similar modes of transport like a blue bus and a red bus: people unmeasured characteristics that make them more likely to choose blue bus may also make them likely to choose red bus because both are similar alternatives (e.g. maybe among the unmeasured characteristics is a preference for public transportation that affects both blue and red bus choice).