A Batch-Size Compound Poisson Desk Flow

Trades arrive according to $N\sim\mathrm{Poisson}(\lambda)$. Each trade contributes a batch size $B$ taking values $0,1,2$ with probabilities $1/2,1/3,1/6$, independently across trades and from $N$. Let \[ S=\sum_{k=1}^N B_k. \] Find the MGF of $S$, and compute $E[S]$ and $\mathrm{Var}(S)$.