What happens to the current at any part of a series circuit?

The current at any part of the series circuit is the same. That is irrespective of where we measure the current in a series circuit. It will be same everywhere or at any point in a given series circuit. In other words,

I = I1 = I2 = I3 = I4........

Where I stands for current and 1, 2, 3, 4, etc. are locations within a series circuit where the current has been measured. 

One can think of the current as the rate of flow of charges through a circuit. In the case of a series circuit, as the charges flow through various devices or resistors, they neither accumulate nor are used up or lost and hence their rate of flow is the same everywhere and that is why the current in any part of the series circuit is the same. 

Hope this helps.