SUMMARY Full-waveform inversion (FWI) is an effective method for imaging subsurface properties using sparsely recorded data. It involves solving a wave propagation problem to estimate model parameters that accurately reproduce the data. Recent trends in FWI have seen a renewed interest in extended methodologies, among which source extension methods leveraging reconstructed wavefields to solve penalty or augmented Lagrangian (AL) formulations have emerged as robust algorithms, even for inaccurate initial models. Despite their demonstrated robustness on synthetic data, challenges remain, such as the lack of a clear physical interpretation and reliance on difficult-to-compute least-squares (LS) wavefields. Moreover, the literature lacks a general and through comparison of these source extension methods with each other and with the standard FWI. This paper is divided into three critical parts. In the first, a novel formulation of these methods is explored within a unified Lagrangian framework. This novel perspective permits the introduction of alternative algorithms that use LS multipliers instead of wavefields. These multiplier-oriented variants appear as regularizations of the standard FWI, are suitable to the time domain, offer tangible physical interpretations, and foster enhanced convergence efficiency. The second part of the paper delves into understanding the underlying mechanisms of these techniques. This is achieved by solving the associated non-linear equations using iterative linearization and inverse scattering methods. The paper provides insight into the role and significance of Lagrange multipliers in enhancing the linearization of the equations. It explains how different methods estimate multipliers or make approximations to increase computing efficiency. Additionally, it presents a new physical understanding of the Lagrange multiplier used in the AL method, highlighting how important it is for improving algorithm performance when compared to penalty methods. In the final section, the paper presents numerical examples that compare different methods within a unified iterative algorithm, utilizing benchmark Marmousi and 2004 BP salt models.