Using　any　nonnegative　function　with　a　nonpositive　derivative　along　trajectories　to　define　a　virtual　output,　the　classic　LaSalle　invariance　principle　can　be　extended　to　switched　nonlinear　time-varying　(NLTV)　systems,　by　considering　the　weak　observability　associated　with　this　output.　The　concept　of　weak　observability　reflects　the　information　of　the　limiting　behavior　of　state　trajectories　hidden　in　the　zero　locus　of　the　output.　In　the　context　of　switched　NLTV　systems,　weak　observability　can　be　explored　using　the　recently　established　framework　of　limiting　zeroing-output　solutions.　Adding　to　this,　a　new　approach　to　show　uniform　global　attractivity　of　a　closed　set　(without　assuming　uniform　Lyapunov　stability　or　dwell-time　conditions)　is　proposed.　A　further　extension　of　the　generalized　LaSalle　invariance　principle　allows　the　consideration　of　cascaded　switched　NLTV　systems,　which　is　used　to　characterize　how　consensus　in　a　leaderless　swarm　of　nonholonomic　robots　allowing　for　switching　communication　topologies　may　be　established.