In Jost's Riemannian Geometry and Geometric Analysis, for proving the existence of geodesic , we consider a heat flow 1.6.2 and 1.6.3
How to get the last two lines? This is integrate , the normal coordinate only can used in a point not a neiborghood . $$ \int -2g_{ij}u^i_{ss}u_t^j-2g_{ij,k}u^k_su^i_tu^j_s+g_{ij,k}u_t^ku^i_su^j_s \\ =\int -2g_{ij}u^i_{ss}u_t^j-g_{ij,k}u^k_su^i_tu^j_s $$
Normal coordinates are not being used, as should be clear by the presence of metric derivatives. Just substitute $u^i_{ss} = u^i_{t} - \Gamma^i_{jk} u^j_s u^k_s$ from the original PDE and use the coordinate expression for the Christoffel symbols in terms of the derivatives of the metric.
I'd warn you that the last part of Jost's proof (Step 5) is incorrect - here he does integrate by parts while using normal coordinates, so he forgets a curvature term. You actually need $M$ to have non-positive curvature for this result (the strong convergence to a geodesic as $t \to \infty$) to hold - otherwise there are counterexamples, see e.g. this paper.