There are many methods. A nice introduction to the various issues involved (for binary choices) is: http://fmwww.bc.edu/EC-C/S2013/823/baum.san2012.pdf

Most of the exogenous variables in econometric models may be functions of other variables in the nature (but out of the model's boarder). Don't worry about that. But if an endogenous variable is dependent of another endogenous variable inside the boundary of your model, then a system (vector/state-space) model should be designed and estimated. No difference, is it a logit, a probit, an AR, an ECM or ... the interactions should be taken into consideration. 

A good contribution on R :

http://www.r-bloggers.com/probit-models-with-endogeneity/

In addition to the helpful references above, you might take a look at Guevara and Ben-Akiva's work, for example:

http://www.icmconference.org.uk/index.php/icmc/ICMC2013/paper/viewFile/521/281

or

Guevara and Ben-Akiva, "Addressing Endogeneity in Discrete Choice Models: Assessing Control-Function and Latent-Variable Methods"

One of the simpler ways to attack this problem is to estimate a bivariate probit model that considers a single endogenous regressor in the first equation.  See section 4.2 in 

http://rt.uits.iu.edu/visualization/analytics/docs/cdvm-docs/cdvm.pdf

for a discussion and some good examples.  I estimate the ordered bivariate probit model in Hamilton, Richards and Stiegert, which is available at researchgate.

You can also estimate it using a two-stage procedure and instrumental variables. Let's call the endogenous explanatory variable Z and the dependent variable in your model Y. First, you find variables that are correlated to Z, but incorrelated to Y. You run an auxiliary regression using these variables as explanatory variables and Z as the endogenous variable. Then, you calculate the estimated values of Z in the auxiliary regression (let's call them Z(e) ). Now, you can estimate your main equation, using Z(e) instead of Z. This new equation should have no endogeneity problems, and the estimates are unbiased and consistent (although they are not efficient).