Appropriate estimator for FDI "gravity" model
An alternative is to use stocks rather than flows, but this does not
  seem to fit very well with the theory of gravity equations which seem
  to me to inherently deal with flow concepts.


Using stocks rather than flows and avoiding the disinvestment issue is a good alternative. Gravity equations have been shown to do a good job fitting stocks of FDI (see the Handbook of International Economics, Gravity Equations: Workhorse, Toolkit, and Cookbook, page 149). 

This paper considers a model in which FDI takes the form of acquisitions. Using the discrete choice framework in a way that resembles Eaton and Kortum (2002), the authors develop a gravity equation for FDI which fits the data well. This paper extends the model to greenfield investment by imagining that instead of bidding for assets, each corporation selects the best "investment project" across all host countries.