  Within the standard weak-coupling limit, the reduced dynamics of open quantum
spin chains with their two end spins coupled to two distinct heat baths at
different temperatures are mainly derived using the so-called \textit{global}
and \textit{local} approaches, in which, respectively, the spin
self-interaction is and is not taken into account. In order to compare the
differences between the two regimes, we concentrate on an open three-site $XX$
spin-chain, provide systematic techniques to addressing the global and local
asymptotic states and then compare the asymptotic spin-transport features by
studying the spin flux through the middle site. %we examine the transport
properties of an open three-site $XX$ spin-chain. %We analytically derive the
exact steady state of the open chain in the global approach and its first order
expansion with respect to the self-interaction in the local one. %The steady
state transport properties through the middle spin is then studied %in order to
compare the physical scenarios associated with the two regimes. %By analytical
and numerical means, Basing on the analytical expressions of the stationary
states in the two regimes, we find that the local approach misses important
global effects emerging as spin sink and source terms that can only be due to
non negligible inter-spin interactions. Moreover, we show that the local
asympotic transport features cannot be recovered from the global ones by
letting the inter-spin coupling vanish thus pointing to the existence of
different coupling ranges where only one approach is physically tenable and
possibly a region where the two descriptions may coexist.
