Let $X_1,X_2,...$ be a discrete-time stochastic process with a distribution $P_θ$, $θ\inΘ$, where $Θ$ is an open subset of the real line. We consider the problem of testing a simple hypothesis $H_0:$ $θ=θ_0$ versus a composite alternative $H_1:$ $θ>θ_0$, where $θ_0\inΘ$ is some fixed point. The main goal of this article is to characterize the structure of locally most powerful sequential tests in this problem. For any sequential test $(ψ,φ)$ with a (randomized) stopping rule $ψ$ and a (randomized) decision rule $φ$ let $α(ψ,φ)$ be the type I error probability, $\dot β_0(ψ,φ)$ the derivative, at $θ=θ_0$, of the power function, and $\mathscr N(ψ)$ an average sample number of the test $(ψ,φ)$. Then we are concerned with the problem of maximizing $\dot β_0(ψ,φ)$ in the class of all sequential tests such that $$ α(ψ,φ)\leq α\quad{and}\quad \mathscr N(ψ)\leq \mathscr N, $$ where $α\in[0,1]$ and $\mathscr N\geq 1$ are some restrictions. It is supposed that $\mathscr N(ψ)$ is calculated under some fixed (not necessarily coinciding with one of $P_θ$) distribution of the process $X_1,X_2...$. The structure of optimal sequential tests is characterized.