Behavior of non-deterministic Turing Machines after $t_i$ seconds
There are no seconds of execution, you might want to use steps instead.
Non-deterministic Turing Machine does not behave randomly at all, so two machines $\mathcal M_1$ and $\mathcal M_2$ (being copy) are identical, they do not in any random way pick the steps, so the simulation on the first one goes like on the second one. At any time after the same number of steps the tape content matches and the current state matches.

In sny transition there might be more than one rule, the NDTM accepts when one of the paths accepts, but these are executed in copied machines. This effectively may mean that the NDTM have "guessed" the next step (counting steps to the accepted path is the shortest possible one if such exist) but it will always be the same one for given configuration. Please note that if you imagine every step as producing copy of the machine with a different transition taken then these copies might differ.

Please read also the difference between non-determinism and randomness