Most English translations seem to render this as "moving things". I asked a native Hebrew-speaking Israeli, and he said the word normally means something on the ground that you would step on (?). Google Translate translates it to "insect", "creeping thing", or "reptile". The Septuagint translates it into Greek as herpeton, which seems to be reptile but seems like it could apply to "creeping things" in general. Wiktionary says it means "small crawling insect".

But it definitely doesn't seem like it applies to all animals, since it is singled out as a separate form of life, apparently, in Genesis 1:25, Genesis 6:7, Genesis 6:20, and others. In Ezekiel 8:10 it is used alongside "abominable beasts" as those things that were depicted in the temple.

In the post-flood dietary restrictions, it also says that you shall not eat flesh with its life blood. One idea I've had is that maybe רמש does insect, since insects would be considered "creeping things", and they don't have blood like other animals (no hemoglobin). But this is just an idea. What do you think?