Does covariance nonstationarity violate GARCH modelling assumptions? I am trying to model two series of financial returns using a bivariate BEKK GARCH estimation.
From the output of the statistical software, I see that covariance stationarity is not ensured in my case ($\alpha_1^2+\beta_1^2 > 1$). That is, either mean, variance or covariance are not constant through time.
Is one of the GARCH modeling assumptions violated then?
Could I also infer that the two processes comove and have a joint trend?
How can I reconcile this result with the fact that correlation between the two time series is almost flat during the time period?
 A: 
Is one of the GARCH modeling assumptions violated then?

What is the main reason you care about model assumptions here? Is it the validity (or lack thereof) of estimation, or something else? I do not immediately see a reason why the estimation of the BEKK-GARCH model would fail (by which I mean e.g. estimators being inconsistent) if the process has a non-stationary conditional variance matrix -- but I may be shortsighted. Anyway, you might still dislike the interpretation of such a model. 

Could I also infer that the two processes comove and have a joint trend?

You need a model for the conditional mean then. If you are modelling returns and the conditional mean model is just a constant for each series, and you find that the (time-varying) conditional correlations are positive, you could say the returns are co-moving.
If you care about a joint stochastic trend in asset prices (rather than their returns), you could try testing for cointegration. If you end up finding the presence of cointegration, the conditional mean specification for the returns would be a vector error correction model (VECM). The conditional variance matrix could still be modelled as a BEKK-GARCH, for example.

How can I reconcile this result with the fact that correlation between the two time series is almost flat during the time period?

Assuming that the model is reasonably well specified (so that the result is not a mistake), you should look for subject-matter explanations for the correlation being almost constant. Perhaps the assets you consider are fundamentally similar (or attract similar investors), and there is then a natural reason for correlation in returns. Or maybe the general market trend is the main driver, so that prices of most of the assets in that market rise and fall together, with a similar degree of correlation (although you might question whether the rise and the fall are symmetric; during falls, the correlations may increase). 
On the other hand, some of the models may just be too rigid to accommodate richer patterns, so that you would not expect large variations in the conditional correlation regardless of the data you have. I wonder how flexible or rigid the BEKK-GARCH is. Are the constraints on parameters (there are some, if I am not mistaken) very restrictive? You could perhaps try generating a series with highly variable conditional correlation and see how well a BEKK-GARCH model captures that.
