The Central Limit Theorem states that the distribution of sample means will form a normal distribution with mean close to the population mean ($\mu$) and variance close to $\sigma^2/n$ for $n$ samples.
However, if you have $n$ samples, you could just treat them all as one large sample and calculate the sample mean and variance from that. The equations for sample mean / variance work with replacement, so it doesn't matter if the samples have overlap.
What real-life situations exist where you would have multiple samples, but not the individuals / draws that make up those samples? Why not just make one large sample? Is the CLT used to derive the sample mean / variance equations?