  Motivated by the study of dynamics of interacting spins for infinite particle
systems, we consider an infinite family of first order differential equations
in a Euclidean space, parameterized by elements $x$ of a fixed countable set.
We suppose that the system is row-finite, that is, the right-hand side of the
$x$-equation depends on a finite but in general unbounded number $n_x$ of
variables. Under certain dissipativity-type conditions on the right-hand side
and a bound on the growth of $n_x$, we show the existence of the solutions with
infinite life-time, and prove that they live in an increasing scale of Banach
spaces. For this, we obtain uniform estimates for solutions to approximating
finite systems using a version of Ovsyannikov's method for linear systems in a
scale of Banach spaces. As a by-product, we develop an infinite-time
generalization of the Ovsyannikov method.
