Low-cost estimation of stationary signals and reduced-complexity tracking of nonstationary processes are well motivated tasks than can be accomplished using ad hoc wireless sensor networks (WSNs). To this end, a fully distributed least mean-square (D-LMS) algorithm is developed in this paper, in which sensors exchange messages with single-hop neighbors to consent on the network-wide estimates adaptively. The novel approach does not require a Hamiltonian cycle or a special bridge subset of sensors, while communications among sensors are allowed to be noisy. A mean-square error (MSE) performance analysis of D-LMS is conducted in the presence of a time-varying parameter vector, which adheres to a first-order autoregressive model. For sensor observations that are related to the parameter vector of interest via a linear Gaussian model and after adopting simplifying independence assumptions, exact closed-form expressions are derived for the global and sensor-level MSE evolution as well as its steady-state (s.s.) values. Mean and MSE-sense stability of D-LMS are also established. Interestingly, extensive numerical tests demonstrate that for small step-sizes the results accurately extend to the pragmatic setting whereby sensors acquire temporally correlated, not necessarily Gaussian data.