form a process corresponding to . However, if we are in a more general setting and deal with integral funnels, their *autonomisation* is not clear. Thus, in the next section we present the precise statement and the proof of the existence of a measurable process. After that, in Section \[loc-proc\] we study the existence of *local processes*. These apply to the situations, where in addition to non-uniqueness we allow solutions to blow up in finite time. \[In the 1960–70s there was some interest in abstracting the dynamical and semi-dynamical systems and processes, see, e.g., [@Hajek-1; @Hajek-2; @B-H]. Our approach involves funnels and is different.\] An example on the semi-process selection is presented in Section \[sec:ex\].

A different and important point of view on non-autonomous dynamics involves the skew-product construction, see, e.g., [@S-S; @S]. The skew-product set-up is very convenient for the study of the long term behavior of non-autonomous systems. However, for the measurable processes selection, we believe our direct approach is more natural. We should mention that our abstract results apply not only to non-autonomous ordinary differential equations, but to partial differential equations, to differential and difference inclusions, and to other situations where integral funnels make sense.

Global processes 