Liquid propellant flow rate control

For a liquid engine, how are mass flow rates from the liquids maintained?

For example in a methane-lox engine, I understand that you can control the pressure output from a regulator, but then how are mass flow rates controlled or measured during launch?

Are mass flow rates measured before ignition and only necessary initially? Are orifice style flow meters and cavitating venturies the solution? If orifice style mass regulators are used, how are the discharge coefficients measured? Or have discharge coefficients been documented reliably enough for different orifice styles?