Assuming that you are asking about mammalian cancer cell lines and ignoring the type of transfection required to get the DNA inside the cell, the plasmid enters the nucleus during the prometaphase where the nuclear membrane gets disorganized. Then depending on the design of your construct, the DNA gets inserted by a single or double crossover mechanism. For stable integrations, double crossovers are preferred. Here, you will have to engineer the flanking regions of your point of insertion into the construct enclosing your desired insertion sequence.
Alternatievly, you can youtube single and double crossovers to get a simple animation to explain the process. Hope this helps.
1. Microinjection: Where the DNA is delivered directly into the nuclei. So, no transportation there.
2. in vivo gene delivery methods (Viral Transduction): Where the viral machinery is exploited to trick the host genome into transcibing and translating your DNA. Virus-host interactions mediate the integration here.
Check out this bitsizebio article: https://bitesizebio.com/9288/principles-and-mechanisms-of-mammalian-cell-transfection/
3. Non-viral vectors: The naked/plasmid DNA enter the nucleus either upon the mitotic disassembly of the nuclear envelope or through nuclear pore complexes in the absence of cell division. This is predominantly mictotubule mediated.
Read more here: Article Cytoplasmic transport and nuclear import of plasmid DNA