So, first note, I’m a biochemist, and who you really need is a historian and a public health major, preferably in the same person. So take the following with a grain of salt.
First off, by ‘raw’, do you mean fresh, or actually raw? Because for the last several millenia humans have cooked lots to most food, since before the dawn of agriculture. So I think you mean fresh and will answer with that. If you mean raw, well, you can probably get lots of that info from people who are on raw diets, but it’s not been a staple of human diets for a very, very long time.
Anyway, it’s more about hygiene then anything else. Depending on the time period, they’d also likely find things super bland depending on their home culture. Most spices we use today were only recently (relatively) made available for common consumption. For instance, only a few centuries ago, pepper was valued as highly as gold by weight. So, that will throw them off. Won’t make it very dangerous, though.
In general, ‘processed’ refers to foods that have been prepared (and usually stabilized somehow) prior to being bought. They may be semi-prepared (say, a boxed meal kit) or fully prepared (like a frozen meal). Different things go into the preparation, but most of the things are either sterilized (which may be as easy as pasteurization), and then stabilizers (say, something to keep it dry). Most of these are quite harmless and rather inert to us. If they weren’t the FDA (or equivalent elsewhere, at least in developed nations) wouldn’t allow them in food. Do stray things get through sometimes? Yes (see also: margarine yellow), but these are the exceptions, not the rule.
So, going back to pre-processed food, you’d definitely notice a difference in taste, and the same vice versa. Either direction would probably have a bit of digestion issues, but more from simple dietary shifts, not really processed foods, and would lead mostly to things like bad gas for a while. Think of a vegetarian going back to meat: if you jump right in your digestion will be messed up for a bit (and going from vegan to carnivore can actually cause pain and damage if you go to fast), but your system will adjust just fine eventually. We eat, say, more simple sugars and less fiber now, and that shift, more than any processing, will cause issues.
As for hygiene, yes, that would cause issues. Though whether they’d be in any more danger than general inhabitants is hard to tell, it would depend heavily. Sometimes you can have something that’s endemic but harmless to the population, but add in someone new and they get hit very hard. The classic example is smallpox, and malaria acts like this to some extent. Different, as it still causes lots of damage, but if you live to adulthood malaria, while still bad, will not incapacitate in the same way it will to someone not previously exposed. So, if there’s something transmissible in that way that the current population may be immune to, that’s possible. It’s less common than something like diphtheria and typhoid, though, which will simply take people down regardless of previous temporal residence. So, while that’s a danger, it’s a little different than what you’re talking about.