Full explanation you can see here, as much as I understand, for Chrome, Safari, Opera and FireFox syntax is:

-webkit-transition

for Internet Explorer syntax is:

transition

I will show example just for Chrome.

HTML:

<!DOCTYPE html>
<html>
<head lang="en">
    <meta charset="UTF-8">
    <title></title>
    <link rel="stylesheet" type="text/css" href="/index.css">
</head>
<body>

    <div id="parentDiv">
        I belong to the parent
    </div>

</body>
</html>

CSS:

#parentDiv {
    border-style: solid;
    border-width: 5px;
    height: 100px;
    width: 100px;
    -webkit-transition: height 15s;
}

#parentDiv:hover {
    height: 8000px;
}

Notice line:

-webkit-transition: height 15s;

here we said that we want to see transition for height, and line:

#parentDiv:hover

Here we are changing height when we come with mouse over.

Example:

I belong to the parent

---

Event which we can use is "webkitTransitionEnd".
For example, HTML:
<!DOCTYPE html>
<html>

	<head>
		<link rel="stylesheet" type="text/css" href="index.css">
		
		<script type="text/javascript" src="jquery-2.1.3.js"></script>
		<script type="text/javascript" src="index.js"></script>
	</head>
	
	<body>
	
		<div id="main">

			<div id="maxHeightDiv">
				Main div with max height
				<div id="nestedDiv">
					Div which is going to move
				</div>		
			</div>

		</div>
		
	</body>

</html>
CSS:
div {
	overflow-y: auto;
}
#maxHeightDiv {
	
	min-height: 100px;
	max-height: 100px;	
	width: 100px;
	overflow-y: auto;
	padding: 60px 0 0 0;
}

#nestedDiv {
overflow-y: auto;
	border: 2px solid #a1a1a1;
	height: 40px;

	-webkit-transition: height 2s;
}

#nestedDiv:hover{
	height: 100px;

	-webkit-transition: height 2s;
}

#maxHeightDiv::-webkit-scrollbar {
	display: none;
}
JS:
function myScroll() {
	$('#maxHeightDiv').scrollTop(10000);
}

window.onload = function () {
	$("#nestedDiv").on("webkitTransitionEnd", function () {
		myScroll();
	});
}
Hover with mouse over "Div which is going to move", and wait a little bit. Real life example:
Main div with max height
Div which is going to move

As you can see scroll will happen only when transition is over.