  A family of continuous-state branching processes with immigration are
constructed as the solution flow of a stochastic equation system driven by
time-space noises. The family can be regarded as an inhomogeneous increasing
path-valued branching process with immigration. Two nonlocal branching
immigration superprocesses can be defined from the flow. We identify explicitly
the branching and immigration mechanisms of those processes. The results
provide new perspectives into the tree-valued Markov processes of Aldous and
Pitman [Ann. Inst. Henri Poincar\'{e} Probab. Stat. 34 (1998) 637-686] and
Abraham and Delmas [Ann. Probab. 40 (2012) 1167-1211].
