  We study the stability behavior of the Bishop-Phelps-Bollob\'as property for
Lipschitz maps (Lip-BPB property). This property is a Lipschitz version of the
classical Bishop-Phelps-Bollob\'as property and deals with the possibility of
approximating a Lipschitz map that almost attains its (Lipschitz) norm at a
pair of distinct points by a Lipschitz map attaining its norm at a pair of
distinct points (relatively) very closed to the previous one. We first study
the stability of this property under the (metric) sum of the domain spaces.
Next, we study when it is possible to pass the Lip-BPB property from scalar
functions to some vector-valued maps, getting some positive results related to
the notions of $\Gamma$-flat operators and $ACK$ structure.
  We get sharper results for the case of Lipschitz compact maps. The behaviour
of the property with respect to absolute sums of the target space is also
studied. We also get results similar to the above ones about the density of
strongly norm attaining Lipschitz maps and of Lipschitz compact maps.
