While quantities conditioned to an isosurface of reaction progress variable c, which characterizes fluid state in a turbulent reacting flow, have been attracting rapidly growing interest in the recent literature, a mathematical and physical framework required for research into such quantities has not yet been elaborated properly. This paper aims at filling two fundamental gaps in this area, i.e., (i) ambiguities associated with a definition of a surface-averaged quantity and (ii) the lack of rigorous equations that describe evolutions of such quantities. In the first (theoretical) part of the paper, (a) analytical relations between differently defined (area-weighted and unweighted) surface-averaged quantities are obtained and differences between them (quantities) are discussed, (b) a unified method for deriving an evolution equation for bulk area-weighted surface-averaged value of a local characteristic ϕ of a turbulent reacting flow is developed, and (c) the method is applied for deriving evolution equations for the bulk area-weighted surface-averaged reaction-surface density |∇c|, local reaction-wave thickness 1/|∇c|, and local displacement speed S_{d}, i.e., the speed of an isosurface of the c(x,t) field with respect to the local flow. In the second (numerical) part of the paper, direct numerical simulation data obtained recently from a highly turbulent reaction wave are analyzed in order to (1) highlight substantial differences between area-weighted and unweighted surface-averaged quantities and (2) show that various terms in the derived evolution equations are amenable to accurate numerical evaluation in spite of appearance of the so-called zero-gradient points [C. H. Gibson, Phys. Fluids 11, 2305 (1968)PFLDAS0031-917110.1063/1.1691820] in a highly turbulent medium. Finally, the obtained analytical and numerical results are used to shed light on the paradox of local flame thinning and broadening which is widely discussed in the turbulent combustion literature.