In eddy current (EC) testing applications, ECs σE (E : electric field, σ: conductivity) are induced in tested metal parts by a low-frequency (LF) source idealized as a closed current loop in air. In the case of highly conducting (HC) parts, a boundary integral equation (BIE) of the first kind under the magneto-quasi-static approximation - which neglects the displacement current - was shown in a previous work to coincide with the leading order of an asymptotic expansion of the Maxwell BIE in a small parameter reflecting both LF and HC assumptions. The main goal of this work is to generalize the latter approach by establishing a unified asymptotic framework that is applicable to configurations that may involve multiple moderately-conducting (σ = O(1)) and non-conducting objects in addition to (possibly multiply-connected) HC objects. Leading-order approximations of the quantities relevant to EC testing, in particular the impedance variation, are then found to be computable from a reduced set of primary unknowns (three on HC objects and two on other objects, instead of four per object for the Maxwell problem). Moreover, when applied to the Maxwell BIE, the scalings suggested by the asymptotic approach stabilize the condition number at low frequencies and remove the low-frequency breakdown effect. The established asymptotic properties are confirmed on 3D numerical examples for simple geometries as well as two EC testing configurations, namely a classical benchmark and a steam generator tube featured in pressurized water reactors of nuclear power plants.