A larger capacitor will be able to supply a larger current but it can not respond as fast. Capacitors are not pure capacitors. They have resistance and inductance. And there are different types with different characteristics and uses. The more you know the more you will find that the answers are not so simple. It comes down to compromises.
A lower value part will typically have a higher self resonant frequency but it also has less capacitance and not as much charge that it can deliver. Sometimes many smaller value capacitors are used rather than a larger one.
On my boards I use a 68uF tantalum capacitor along with a 10uf ceramics, plus a couple of 0.1uF ceramics. The parts I use are surface mount parts that are mounted right at the ESP module. But I also have a 500mA low drop regulator mounted at the modules. Some people say they have had good results with 470uF. I find them to be bigger than I would like. The parts I use are relatively small.
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There is a regulator, a 68uF and 10uF cap in the lower left corner of the ESP07 module. Right by the power input tabs of the module.
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A couple of years ago I made some small PCBs that included some parts along with the ESP12 modules. Cleaner than the above but basically the same thing.
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I picked 68uF because that is what I had at work. Same for that 10uF. The actual value is only part of the solution and not really a critical one. The right type of parts and the implementation is more important.
This is a board I designed for work. Same idea. Same cap right at the power input of the module.
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