  We investigate the relativistic properties under boost transformations of the
$\kappa$-Poincar\'e model with multiple causally connected interactions, both
at the level of its formulation in momentum space only and when it is endowed
with a full phase space construction, provided by the relative locality
framework. Previous studies focussing on the momentum space picture showed that
in presence of just one interaction vertex the model is relativistic, provided
that the boost parameter acting on each given particle receives a
"backreaction" from the momenta of the other particles that participate in the
interaction. Here we show that in presence of multiple causally-connected
vertices the model is relativistic if the boost parameter acting on each given
particle receives a backreaction from the total momentum of all the particles
that are causally connected, even those that do not directly enter the vertex.
The relative locality framework constructs spacetime by defining a set of dual
coordinates to the momentum of each particle and interaction coordinates as
Lagrange multipliers that enforce momentum conservation at interaction events.
We show that the picture is Lorentz invariant if one uses an appropriate "total
boost" to act on the particles' worldlines and on the interaction coordinates.
The picture we develop also allows for a reinterpretation of the backreaction
as the manifestation of the "total boost" action. Our findings provide the
basis to consistently define distant relatively boosted observers in the
relative locality framework.
