
During mitosis, where do the organelles go as the process is occurring?

To answer your exact question, I have to expand the topic a bit and include what happens right after mitosis: cytokinesis, which by definition describes the division of the cytoplasm and its contents. When we consider the possibilities for what might happen during mitosis, we can limit what can happen to the organelles: stay put unchanged or stay put but temporarily transform. Cytokinesis is likely more complicated; cells have many types of organelles, and those organelles do and can do many things. I think it is safe to assume that this variety will likely implicate several mechanisms of organelle division among the daughter cells. Whatever the mechanisms of these processes, they will fall into two main categories: active processes requiring dedicated structures and activities and passive processes in which organelles are simply divided by the all-encompassing cell membrane as it splits in two. I begin the discussion with the cell nucleus and the mitochondria because they perform the crucial functions of storing the protocols and powering the cellular factory, respectively. We know that the nuclear envelope breaks down during mitosis before the chromosomes align along the nuclear spindle, which is composed of microtubules. This structure ensures the equal division of chromosomes between the two daughter cells and serves as an example of an active process. Mitochondria are a bit of a curiosity (see Daniel Pennington’s Q&A to find out why), but they also actively divide by forming dedicated machinery. In general, division or partitioning of organelles occurs more or less randomly from a parent cell; this is crucial because cells cannot form membrane-bounded organelles from scratch.
The dedicated functions of cell organelles are further reflected by the mechanisms of inheritance of these organelles by the daughter cells after mitosis. Protocols in the form of mRNA from the cell nucleus are sent to the endoplasmic reticulum (ER) for translation into proteins; these proteins are further modified by the Golgi apparatus. Interestingly enough, it seems that the ER and Golgi may be inherited in association with the nucleus and the mitotic spindle. In contrast, lysosomes and vacuoles perform lytic functions and serve as reservoirs for ions and metabolites. These functions are intimately associated with the energy conversion processes that occur in the mitochondria. Once again, it has been shown that the mitochondria are required for the inheritance of lysosomes and vacuoles; these organelles are carried on the mitochondrial spindle during cell division.
To the best of my knowledge, this summary samples the current knowledge of the inheritance of organelles during and after mitosis. I find it fascinating that the inheritance of organelles is not purely random but rather elegantly is prioritized by their required functions. Intuitively, this seems reasonable as it is crucial to ensure that cellular machinery is inherited in functional packages. Imagine how difficult it would be to build a car if, rather than receiving the parts for the engine in a single shipment, the components were sent to you at random and mixed with parts for the dashboard and suspension! The inherited packages of organelles efficiently perform the task of quality control.
Thank you for your question; I learned a fair amount from researching the topic, and I hope you find it empowering that perhaps the most powerful knowledge or skill you can obtain is the ability for lifelong learning. Anything that you don’t know you can discover!
References:
1. Symens N, Soenen SJ, Rejman J, Braeckmans K, De Smedt SC, Remaut K. Intracellular partitioning of cell organelles and extraneous nanoparticles during mitosis. Adv Drug Deliv Rev. 2011.
2. Imoto Y, Yoshida Y, Yagisawa F, Kuroiwa H, Kuroiwa T. The cell cycle, including the mitotic cycle and organelle division cycles, as revealed by cytological observations. J Electron Microsc (Tokyo). 2011; 60(suppl 1):S117–S36.