How would I check the validity of covariates in my linear model on several hundred datasets?

I have this linear model with predictors that I need to prove are statistically significant and pass the necessary lm assumptions. I know for a single dataset, I can use various LM tests, but the problem is I have several hundred datasets which cannot be combined. Coefficients may be different for each dataset, but I just need to prove(or disprove) that the covariates can be used for lm across all models. I'm assuming I shouldn't run tests on each LM one by one.

How should I approach this problem?