If I have three button cells in series, and one of them dies, will the whole circuit still function at a lower voltage or will it stop flowing?
The circuit will continue to function at reduced voltage. Note that the voltage declines over time. It does not suddenly drop to zero. Normally, you want to remove the load from the batteries before any one cell drops to zero volts, because the other two batteries will force the dead one to discharge to negative cell voltage. This "reversed" cell may then leak electrolyte in the battery compartment (depending on chemistry and construction details). If you plan to do this, you should consult with the cell vendor to make sure it will not cause leakage or other problems.

Most of my experience is with 1.5V alkaline cells and rechargeable Lithium polymer cells, not with button cells.

You can also test it by putting a discharged cell in series with two strong cells, then discharge the series stack until the weak cell reverses.