Generally speaking, the responsibility of proof falls on party making the claim.

If I claim that you beat your wife, you are not expected to prove your innocence by showing that you don't do it. Proving a negative is difficult if not impossible in some cases. I have to show evidence to back up my claim.

It's a big claim that "immigrants are eating the cats and dogs in specific town in Ohio" (note the plural).

What was readily checked is the source of such a claim (where did Trump get that from?) and what evidence was provided?

The trace back on that stupidity was unsubstantiated rumours triggered from a walked back local area posting and a slew of images that didn't come from the place in question, etc.