  We study twisted cohomologies with paracompactifying families of supports.
The Kunneth theorems, Leray-Hirsch theorems and self-intersection formulae are
established. Based on these results, we eventually give explicit expressions of
complex blow-up formulae for twisted Dolbeault cohomology on arbitrary complex
manifolds and the ones of generalized blow-ups formulae for twisted de Rham
cohomology on arbitrary oriented smooth manifolds. These expressions are
induced by the morphisms of (simple or double) complexes of spaces of forms and
currents rather than just the maps between cohomologies, which help us to
obtain the corresponding results for twisted Bott-Chern, Aeppli cohomologies
and hypercohomologies of truncated twisted holomorphic de Rham complexes.
