For any given modern turbofan engine, does bypass ratio stay approximately constant during a typical flight?
There are two factors that determine the bypass ratio (or rather, how it may change):


The first is the relative diameter of the fan, compared to the core. Obviously, this is fixed in the design process, and doesn't change during flight
The second is the relative difference in RPM speed between the two (or even 3) shafts that exist in the engine. This does change during a flight, from the takeoff condition, to a lower thrust setting at cruise. So let's look at this in detail.


The compressor acts like a pump, increasing the pressure of the air, whether it's sucking at the intake like in takeoff, or RAM feed, such as in cruise flight. Also like a pump, the faster a spool (compressor) spins, the larger the mass flow it pumps (creates). So, if we consider a CFM56-7B engine for example, its shaft speeds at 100% are 5,175 rpm for N1 and 14,417 rpm for N2. That's a ratio of 2.78. At takeoff, the engine will be close to this condition. But at cruise thrust at 35,000ft, the engine will be throttled back. What tends to happen in a gas turbine, is the N1 spool slows down relatively more than N2, as the throttle is reduced. So at cruise, N1 is maybe 3,880 rpm (75%), while N2 is maybe 11,534 rpm (80%). The ratio between them is now 3.41. The pumping capacity of the core has only reduced 20% from the takeoff condition, while the pumping capacity of the fan has reduced 25%. Hence, the bypass ratio will be lower at cruise than at takeoff.

Now, my values of 80% and 75% at cruise are just suggestions. But it is to illustrate the point, that if the ratio between the rpm of the two spools change, the bypass ratio will change.

In essence, the core flow is being determined by the rpm of both spools, while the fan duct flow is just determined by the rpm of the fan. So changes in the rpm ratio will impact the bypass ratio.

{Why is the N2 operating range less than the N1 range?: The CFM56 engine has a single stage fan, and depending on the model, either a 3 or 4 stage booster (or LPC), with a 9 stage HPC. These compressors are making the air flow in the direction of increasing pressure. The air doesn't really want to do that. So, if the design tries to increase the pressure ratio per stage too much, the blade is likely to stall. Getting aerodynamic conditions which avoid stall for the first stage, and the last stage, for the same operating condition is difficult. It's even harder when there are 9 stages on the shaft, than 3 or 4 (plus a fan). So the range of conditions that a 9 stage compressor will operate over is generally less than the range of conditions that a 3 or 4 stage compressor will operate over.}