Shabbos & Kashrus
A Yid may not instruct a non-Y to perform during Shabbos, the type of work a Yid himself may not perform. However, the non-Y may choose for his own convenience to perform the task during Shabbos.
We may on Friday just prior to Shabbos, instruct a non-Y worker to perform a task that requires a few hours to complete, ready by Monday or even Sunday morning. Although it is quite likely that the non-Y may choose to work on Saturday - as long as there is no instruction implicit or explicit - to perform the task during Shabbos.
We must differentiate however, between the Yid doing what is prohibited and the food becoming prohibited.
A food manufacturing business, fully owned by Yidden, may not operate during Shabbos - if it produced food for the general population. The owners, no matter how distant they may be from the actual management and production team, are in essence, instructing the production for their own benefit.
However, notwithstanding that they have transgressed the Halacha by cooking during Shabbos and Yom Tov, the foods are Kosher.
The reason for this is that the foods are not cooked specifically for Jews.
Even those facilities producing foods bearing a paid for, legitimate, Kosher seal, are permitted because they are not prepared specifically for the Kosher consumers. Batch production bearing a private label such as Leibers, Glicks, Chaim's, Pashkez or Tempo will be prohibited since they are manufactured specifically for and purchased almost exclusively by, Kosher consumers.
Foods produced in Israel are generally not manufactured during Shabbos and YomTov. And even if they might be, exports to the international and sales to local, non-J markets is a significant factor in overall production.