I'm unsure how to interpret the output of a fixed effects (in the econometric sense) logistic regression model using cross-sectional data. I'm struggling with the 'cluster-specific' interpretation these models have. I sort of understand how this works with panel data, but I'm not sure whether the coefficients have the same interpretation with cross-sectional data.

I have archeological data from ~800 individuals, sampled from 6 countries (n=75 to n=200), at 3 time periods ('pre event', 'during event', 'post event'). The 3 time periods are cross-sectional (i.e., different individuals sampled in each) and are not contemporaneous across countries. So, 'pre', 'during', and 'post' occur at different times (sometimes hundreds of years apart) in different countries, but individuals are always sampled before, during, and after a 'shock' that is comparable across populations. My response is a binomial proportion at the individual level. My predictor is also at the individual level and I'd like to model whether it is moderated by time period. My main focus therefore is whether there are changes in the relationship between predictor and response across the 3 time periods. I have no variables at the country level - everything is at the individual level, and I don't really care about the countries per se.

My thinking is that to control for the clustering by country and all the possible unobserved country-level variables I should use a fixed effects model. I've therefore built the following model (in R syntax):

Code: 
glm(cbind(success, total - success) ~ factor(time_period) * predictor1 + factor(country),
          family = binomial(link = "logit"),
          data = dat)
where 'factor(country)' creates 5 dummy variables for 6 countries, 'factor(time_period)' creates 2 dummy variables for 3 time periods, and 'predictor1' is a continuous predictor at the individual level. This gives me a within-cluster model, which in the context of my data means within-country. (Incidental parameter bias shouldn't be a problem with 6 clusters and >75 observations in each - I get almost identical results using a conditional likelihood approach.)

What I don't understand is how this works (or maybe it doesn't) when I have cross-sectional time periods within each country. I'd be grateful for any help in understanding how to interpret the interaction effects - specifically relating to their cluster-specific interpretation. I know that the two interaction terms will tell me about the differences in slope for 'predictor1' between the reference time period 'pre' and the other two time periods. But, are these difference for a 'typical' country, or averaged across countries, or something else?