Is it possible that in your old site, the ordering of the CSS files in the HTML are different?
Let's say that, in your main CSS file (main.css), you set a font size of anything inside the body tag to be 10px, and in a page-specific CS file (thisPage.css), you set the font size of anything in the body tag to be 12px.
<html>
<head>
<link rel="stylesheet" href="main.css"/>
<link rel="stylesheet" href="thisPage.css"/>
</head>
<body>
...
</body>
</html>
In a page rendering, the font size will be 12px. The page-specific CSS will override anything that matches a rule in the main CSS (unless of course, you apply the !important rule!).
<html>
<head>
<link rel="stylesheet" href="thisPage.css"/>
<link rel="stylesheet" href="main.css"/>
</head>
<body>
...
</body>
</html>
In this page rendering, the font size will be 10px.
This may be something to consider as, because these are cascading rules, the order is important.