In this paper, we present the asymptotic theory for integrated functions of increments of Brownian local times in space. Specifically, we determine their first-order limit, along with the asymptotic distribution of the fluctuations. Our key result establishes that a standardized version of our statistic converges stably in law towards a mixed normal distribution. Our contribution builds upon a series of prior works by S. Campese, X. Chen, Y. Hu, W.V. Li, M.B. Markus, D. Nualart and J. Rosen [2, 3, 4, 5, 10, 13, 14], which delved into special cases of the considered problem. Notably, [3, 4, 5, 13, 14] explored quadratic and cubic cases, predominantly utilizing the method of moments technique, Malliavin calculus and Ray-Knight theorems to demonstrate asymptotic mixed normality. Meanwhile, [2] extended the theory to general polynomials under a non-standard centering by exploiting Perkins’ semimartingale representation of local time and the Kailath-Segall formula. In contrast to the methodologies employed in [3, 4, 5, 13], our approach relies on infill limit theory for semimartingales, as formulated in [6, 8]. Notably, we establish the limit theorem for general functions that satisfy mild smoothness and growth conditions. This extends the scope beyond the polynomial cases studied in previous works, providing a more comprehensive understanding of the asymptotic properties of the considered functionals.