Too many vias can make EMI worse
The parastics for vias look like the circuit below. With the vias the same size and making the grid smaller would only reduce the inductance and resistance between vias.





The via sizing will make a difference, making the via smaller (with the height the same increases the inductance. However, there are more vias so over all the inductance drops. The specific frequency does matter, and can be estimated per your application. Since the values are in the uΩ, pF and pH range, the time to care about this is when you are dealing with microwaves. 


Source: https://www.microwaves101.com/encyclopedias/microstrip-via-hole-inductance