The engines will mechanically swap over easily
However the wiring will be different and usually with different plugs
Not a big problem if coloured wires do not figure regularly in your nightmares
The working bits are the same so the wire to the coil on one will go to the coil on the other
The wire to the carb solenoid on one will go to the carb solenoid on the other
The B &S engine will have a remote starter solenoid while the Kohler will probably have a piggy back starter