Aright then I found a solution to download files from Unity which aren’t host on the same domain.
I share it in case it can be improved and help some 
From Unity C# :
[DllImport("__Internal")]
private static extern string DownloadFileFromUrl(
string url,
string objSource,
string msgOnProgress,
string msgOnLoaded,
string msgOnError);
public void Start()
{
DownloadFileFromUrl(
inputUrl.text,
gameObject.name,
"OnDataDownloadProgress",
"OnDataDownloadSuccess",
"OnDataDownloadError");
}
public void OnDataDownloadProgress(float progress)
{
}
public void OnDataDownloadSuccess(string base64)
{
}
public void OnDataDownloadError(string msg)
{
}
From Unity *.jslib :
mergeInto(LibraryManager.library,
{
DownloadFileFromUrl: function (url, objSource, msgOnProgress, msgOnLoaded, msgOnError)
{
AjaxLoaderFromHtml(Pointer_stringify(url),
Pointer_stringify(objSource),
Pointer_stringify(msgOnProgress),
Pointer_stringify(msgOnLoaded),
Pointer_stringify(msgOnError));
},
});
From here to have a custom WebGLTemplates and to include the following external.js script :
(base ajax code by Cyril Pereira : https://med.planet-d.net/)
function NewAjaxLoader(url, objSource, msgOnProgress, msgOnLoaded, msgOnError) {
var xhr = new XMLHttpRequest();
xhr.open("GET", url, true);
xhr.responseType = "arraybuffer";
xhr.onprogress = function (e) {
if (e.lengthComputable) {
var progress = Math.round((e.loaded / e.total) * 100);
console.log("progress : " + String(progress));
unityInstance.SendMessage(objSource, msgOnProgress, progress);
}
};
xhr.onreadystatechange = function () {
if (xhr.readyState === 4) {
var arrayBuffer = xhr.response;
var base64String = btoa(
String.fromCharCode.apply(null, new Uint8Array(arrayBuffer))
);
unityInstance.SendMessage(objSource, msgOnLoaded, base64String);
}
};
xhr.onerror = function (e) {
unityInstance.SendMessage(objSource, msgOnError, String(e));
};
return xhr;
}
Finally from the index.html file :
<script src='external.js'></script>
<script>
function AjaxLoaderFromHtml(url, objSource, msgOnProgress, msgOnLoaded, msgOnError)
{
NewAjaxLoader(url, objSource, msgOnProgress, msgOnLoaded, msgOnError).send();
}
</script>
It’s very important the ajax.send() is made from the *.html if you want to escape the external Javascript request being blocked by webGL.
Voila 
