In testing a global null hypothesis, with independent tests, the p-values are distributed as $U(0,1)$. There are many goodness-of-fit tests to check that the distribution is uniform. For example, this answer discusses the use of chi-squared, Kolmogorov-Smirnov, and several others.
However, in many articles about combining test results, I’ve seen the recommendation to use Fisher’s combined test. Does it have any specific advantage for global null testing, compared to the more widely used tests of uniformity that I listed above?
The problem with the combination of $p$-values is that the question is usually not well specified. The methods available fall into two groups which are more sensitive to either (a) at least one $p_i$ is not from the uniform distribution (b) they all are not. Fisher’s method is quite sensitive in situation (a) whereas Stouffer’s method is sensitive in situation (b).