In this article, we apply a probabilistic approach to study general mean field type control (MFTC) problems with jump-diffusions, and give the first global-in-time solution. We allow the drift coefficient $b$ and the diffusion coefficient $σ$ to nonlinearly depend on the state, distribution and control variables, and both can be unbounded and possibly degenerate; besides, the jump coefficient $γ$ is allowed to be non-constant. To tackle the non-linear and control-dependent diffusion $σ$, we further formulate a joint cone property and estimates for both processes $P$ and $Q$ of the corresponding adjoint process (where $(P,Q,R)$ is the solution triple of the associated adjoint process as a backward stochastic differential equation with jump), in contrast to our previous single cone property of the only process $P$. We study first the system of forward-backward stochastic differential equations (FBSDEs) with jumps arising from the maximum principle, and then the related Jacobian flows, which altogether yield the classical regularity of the value function and thus allow us to show that the value function is the unique classical solution of the HJB integro-partial differential equation. Most importantly, our proposed probabilistic approach can apparently handle the MFTC problem driven by a fairly general process far beyond Brownian motion, in a relatively easier manner than the existing analytic approach.