Step 1: Recall the components of a dry cell.
In a dry cell, zinc acts as the anode, carbon rod is the cathode, and the electrolyte is a paste of $NH_4Cl$ and $ZnCl_2$.
Step 2: Identify the process at the cathode.
At the cathode, reduction takes place. In a dry cell, $NH_4^+$ ions from ammonium chloride take part in the reduction process.
Step 3: Cathodic reaction.
\[
2NH_4^+ + 2e^- \rightarrow 2NH_3 + H_2
\]
Thus, ammonium ions gain electrons and are reduced at the cathode.
Step 4: Conclusion.
The change occurring at the cathode during the working of a dry cell is the reduction of $NH_4^+$ ions.