In this paper, we propose and investigate nonautonomous first-order differential systems governed by monotone operators. They can be regarded as alternative formulations of second-order continuous Newton-like methods (with asymptotically vanishing hyperbolic regularization). Existence, uniqueness, and weak convergence results to equilibria are established in the setting of Hilbert spaces relative to any maximal monotone operator. Significant convergence rates are also obtained, while our systems involve only operator evaluation.