In general, use "Throughout time" and "During the time of" or "During this time". If you want to use "Throughout the time", you will need to qualify it more, such as "Throughout the time
that the subject was studied"
For your examples, it's tough to figure out which option to go with without the context of any surrounding sentences.
"Throughout" typically lasts from the start of something all the way to the end. So to say "I had a job throughout the year" would mean that you were employed in January and stayed at the job every month, including December of that year.
"During" is not quite as comprehensive. It doesn't have to include the entire timespan of something. So using the above example, "I had a job during the year" would mean that at some point in the year, you were employed. It could have been only in March, and no other months. It could mean you had a job for 6 months out of the year. And it could also overlap with "throughout" and mean you had a job from January all the way through December, but that's not guaranteed when you say "during".
Just based on the options you've provided, I would go with "During this time, one can realise plagiarism became more common."
Although in American English, using "one" in this context comes across as formal and stiff. Something like "During this time, plagiarism became more common." or better yet, "During this time, individuals realized plagiarism became increasingly common."