Leaflet draggable Marker

you can use map.panTo():

function onMapClick(e) {
  marker = new L.marker(e.latlng, {draggable:'true'});
  marker.on('dragend', function(event){
    var marker = event.target;
    var position = marker.getLatLng();
    marker.setLatLng(new L.LatLng(position.lat, position.lng),{draggable:'true'});
    map.panTo(new L.LatLng(position.lat, position.lng))
  });
  map.addLayer(marker);
};

map.on('click', onMapClick);

Complete html with inline js (minus cloudmade api):

<link rel="stylesheet" href="http://cdn.leafletjs.com/leaflet-0.7.1/leaflet.css" />
<!--[if lte IE 8]>
 <link rel="stylesheet" href="http://cdn.leafletjs.com/leaflet-0.7.1/leaflet.ie.css" />
<![endif]-->
<script src="http://cdn.leafletjs.com/leaflet-0.7.1/leaflet.js"></script>

<div id="map" style="width: 600px; height: 600px;"></div>

<script>

var map = L.map('map').setView([51.505, -0.09], 13);

L.tileLayer('http://{s}.tile.cloudmade.com/API-KEY/997/256/{z}/{x}/{y}.png', {
  attribution: 'Map data &copy; <a href="http://openstreetmap.org">OpenStreetMap</a> contributors, <a href="http://creativecommons.org/licenses/by-sa/2.0/">CC-BY-SA</a>, Imagery ©     <a href="http://cloudmade.com">CloudMade</a>',
  maxZoom: 18
}).addTo(map);

function onMapClick(e) {
  marker = new L.marker(e.latlng, {draggable:'true'});
  marker.on('dragend', function(event){
    var marker = event.target;
    var position = marker.getLatLng();
    marker.setLatLng(new L.LatLng(position.lat, position.lng),{draggable:'true'});
    map.panTo(new L.LatLng(position.lat, position.lng))
  });
  map.addLayer(marker);
};

map.on('click', onMapClick);

</script>

In case someone else stumbles upon this and needs a live pan, too.

Just remove the click event like so:

var marker = new L.marker([lat,lon],{
    draggable: true
}).addTo(map);

marker.on("drag", function(e) {
    var marker = e.target;
    var position = marker.getLatLng();
    map.panTo(new L.LatLng(position.lat, position.lng));
});

I wished it would be even more real time, it lags a little behind the mouse.

Leaflet now provides this as an option with creating a marker, so there is no need to attach a drag event to pan the map.

Link to docs: https://leafletjs.com/reference-1.3.4.html#marker-autopan

Code sample:

var marker = new L.marker([lat,lon],{
    draggable: true,
    autoPan: true
}).addTo(map);