Is it possible for series connected battery cells to provide different currents?

In a series circuit, the current in each component of the series will be the same. This is true no matter what the component may be: resistor, capacitor, inductor, diode, battery, etc.

I think the key part is the BMS system you mentioned. I believe it refers to charging control circuits that have circuitry to monitor individual cells or sets of cells, and circuitry capable of shunting charging current 'around' cells that are reaching full charge before others. As such, this arrangement is NOT strictly in series, as the additional circuitry provides parallel current paths on each cell (or set of cells). I see this typically when I open Li-ion battery packs.

I don't believe they do any shunting during the discharge cycle (but I could be wrong).

Technically, everyone else stating that if just the batteries alone are connected in series (no balancing circuitry) that the current is the same in all, are also correct.

Assuming no other connection between the batteries, no. This would be a violation of Kirchoff's Current Law. Imagine a node between the batteries. All the current entering the node must leave the node.