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I need a reference with a complete proof for the following theorem:

If $\{X_t; t\in [0,T)\}$ is a continuous time-dependnet family of vector fields on a compact manifold, then there exists a one-parameter family of diffeomorphisms $\{\varphi_t:M\to M; t\in [0,T)\}$, such that $$\partial_t\varphi_t(x)=X_t(\varphi_t), \varphi_0=id,$$ for all $x\in M$ and $t\in [0,T).$

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    You might want to consider the vector field $Y = (X, 1)$ on $M \times I$; the flow for $Y$ will take $X \times \{0\}$ to $X \times \{t\}$ for every $t$, thus providing you with the diffeomorphisms you seek. But I don't know a book that writes out this proof (or any other). – 2017-02-02

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Theorem 4.3 here I believe is what you are looking for. Of course, some earlier results are also needed.

Moreover, this topic is covered in various textbooks like:

  1. Introduction to Smooth Manifolds - Lee
  2. An Introduction to Differentiable Manifolds and Riemannian Geometry - Boothby
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    Lee's chapter on flows (in particular he section on time dependent vector fields) is absolutely phenomenal – 2017-02-02