How to get all markers on Leaflet

Use eachLayer method on L.map. Like

map.eachLayer(function (layer) { 
    if (layer.options.name === 'XXXXX') {
        layer.setLatLng([newLat,newLon])
    } 
});

Documentation at http://leafletjs.com/reference-1.2.0.html#map-eachlayer

To add an option without using map.eachLayer; all layers within the map are internally stored in map._layers.

Use

map._layers.forEach(function (layer) {
  ...
});

to iterate over ALL Layer elements. Not just the ones currently visible.