Well, here is the final answer. I have used great Jimmy idea (which unfortunately is not complete itself) and complete recursion function to work properly.

Based on interface:

string RemoveAllNamespaces(string xmlDocument);

I represent here final clean and universal C# solution for removing XML namespaces:

//Implemented based on interface, not part of algorithm
public static string RemoveAllNamespaces(string xmlDocument)
{
    XElement xmlDocumentWithoutNs = RemoveAllNamespaces(XElement.Parse(xmlDocument));

    return xmlDocumentWithoutNs.ToString();
}

//Core recursion function
 private static XElement RemoveAllNamespaces(XElement xmlDocument)
    {
        if (!xmlDocument.HasElements)
        {
            XElement xElement = new XElement(xmlDocument.Name.LocalName);
            xElement.Value = xmlDocument.Value;

            foreach (XAttribute attribute in xmlDocument.Attributes())
                xElement.Add(attribute);

            return xElement;
        }
        return new XElement(xmlDocument.Name.LocalName, xmlDocument.Elements().Select(el => RemoveAllNamespaces(el)));
    }

It's working 100%, but I have not tested it much so it may not cover some special cases... But it is good base to start.

I guess you'd want something like:

XmlDocument doc = new XmlDocument();
doc.Load("my.xml");
foreach(var node in doc.DocumentElement.ChildNodes)
{
    var el = node as XmlElement;
    if (el != null)
    {
        if (el.HasAttribute("xmlns"))
        {
            var ns = el.GetAttribute("xmlns");
            if (ns != null && ns != el.NamespaceURI)
            {
                el.RemoveAttribute("xmlns");
            }
        }
    }
}
doc.Save("new.xml");

Obviously you'd probably want to make it query, or issue an all-elements query, in complex scenarios.

I think the code below is what you want. You need to put each element into the right namespace, and remove any xmlns='' attributes for the affected elements. The latter part is required as otherwise LINQ to XML basically tries to leave you with an element of

Copy<!-- This would be invalid -->
<Firstelement xmlns="" xmlns="http://my.namespace">

Here's the code:

Copyusing System;
using System.Xml.Linq;

class Test
{
    static void Main()
    {
        XDocument doc = XDocument.Load("test.xml");
        // All elements with an empty namespace...
        foreach (var node in doc.Root.Descendants()
                                .Where(n => n.Name.NamespaceName == ""))
        {
             // Remove the xmlns='' attribute. Note the use of
             // Attributes rather than Attribute, in case the
             // attribute doesn't exist (which it might not if we'd
             // created the document "manually" instead of loading
             // it from a file.)
             node.Attributes("xmlns").Remove();
             // Inherit the parent namespace instead
             node.Name = node.Parent.Name.Namespace + node.Name.LocalName;
        }
        Console.WriteLine(doc); // Or doc.Save(...)
    }
}

I have found a couple of XSLT solutions to do this, both of which can conveniently be processed with xmlstarlet:

1. How to remove the namespace and its prefixes in an XML file using XSLT? - IBM
2. XSLT: Remove namespace prefix from elements - StackOverflow

In my worked example below, I've used the IBM code and saved it into the file xslt_ibm:

<?xml version="1.0"?>
<xsl:stylesheet xmlns:xsl="http://www.w3.org/1999/XSL/Transform" version="1.0">
  <!-- https://www.ibm.com/support/pages/how-remove-namespace-and-its-prefixes-xml-file-using-xslt -->
  <xsl:output method="xml" indent="yes"/>
  <xsl:template match="/|comment()|processing-instruction()">
    <xsl:copy>
      <xsl:apply-templates/>
    </xsl:copy>
  </xsl:template>
  <xsl:template match="*">
    <xsl:element name="{local-name()}">
      <xsl:apply-templates select="@*|node()"/>
    </xsl:element>
  </xsl:template>
  <xsl:template match="@*">
    <xsl:attribute name="{local-name()}">
      <xsl:value-of select="."/>
    </xsl:attribute>
  </xsl:template>
</xsl:stylesheet>

If you consider your XML file to be saved in the file file.xml then this command will rename each element and attribute out of its namespace:

xmlstarlet transform xslt_ibm file.xml

Input (file.xml)

<md:EntityDescriptor xmlns="urn:_" xmlns:md="_"></md:EntityDescriptor>

Output

<?xml version="1.0"?>
<EntityDescriptor/>

I'd like to expand upon the existing answers. Specifically, I'd like to refer to a common use-case for removing namespaces from an XElement, which is: to be able to use Linq queries in the usual way.

When a tag contains a namespace, one has to use this namespace as an XNamespace on every Linq query (as explained in this answer), so that with the OP's xml, it would be:

XNamespace ns = "http://www.blablabla.com/bla";
var element = xelement.Descendants(ns + "retEvent")).Single();

But usually, we don't want to use this namespace every time. So we need to remove it.

Now, @octaviocc's suggestion does remove the namespace attribute from a given element. However, the element name still contains that namespace, so that the usual Linq queries won't work.

Console.WriteLine(xelement.Attributes().Count()); // prints 1
xelement.Attributes().Where( e => e.IsNamespaceDeclaration).Remove();
Console.WriteLine(xelement.Attributes().Count()); // prints 0
Console.WriteLine(xelement.Name.Namespace); // prints "http://www.blablabla.com/bla"
XNamespace ns = "http://www.blablabla.com/bla";
var element1 = xelement.Descendants(ns + "retEvent")).SingleOrDefault(); // works
var element2 = xelement.Descendants("retEvent")).SingleOrDefault();      // returns null

Thus, we need to use @Sam Shiles suggestion, but it can be simplified (no need for recursion):

private static void RemoveAllNamespaces(XElement xElement)
{
    foreach (var node in xElement.DescendantsAndSelf())
    {
        node.Name = node.Name.LocalName;
    }
}

And if one needs to use an XDocument:

private static void RemoveAllNamespaces(XDocument xDoc)
{
    foreach (var node in xDoc.Root.DescendantsAndSelf())
    {
        node.Name = node.Name.LocalName;
    }
}

And now it works:

var element = xelement.Descendants("retEvent")).SingleOrDefault();