- Article history: Consistent Lagrangian description of variable-tension lightlike p-branes (LL-branes )i s presented in two equivalent forms - a Polyakov-type formulation and a dual to it Nambu-Goto-type formulation. An important and non-standard characteristic feature of the LL-branes is that the brane tension appears as a non-trivial additional dynamical degree of freedom. We consider properties of p = 2 LL-brane dynamics (as a test brane) in D = 4 Kerr or Kerr-Newman gravitational backgrounds in some detail. It is shown that the LL-brane automatically positions itself on the horizon and rotates along with the same angular velocity. Finally, we construct explicitly a traversable wormhole of Misner-Wheeler type based on a Reissner- Nordstrom geometry. This wormhole is constructed as a self-consistent solution of the electrically sourceless Einstein-Maxwell system in the D = 4 bulk interacting with a LL-brane .T he pertinent wormhole throat is located precisely at the LL-brane sitting on the outer Reissner-Nordstrom horizon with the Reissner-Nordstrom mass and charge being functions of the dynamical LL-brane tension. © 2009 Published by Elsevier B.V.