Here is my sample code - I am thinking about contributing a module for this function

// get the header code and write it to a file function module_name_global_header_write() {//dpm('raw_html');

    // there may be an issue if we are processing this function while viewing the same url as $sourceDomain
    // perhaps we will need to create a page or URL that is not viewed but can supply the HTML for the header
    $sourceDomain = 'http://site_name.com';
    $sourceMenuId = 'superfish-1';  

    // initialize simplehtmldom objects
    $html = new simple_html_dom();
    $responsive_desktop_css = new simple_html_dom();
    $global_styles_css = new simple_html_dom();

    // load files into simplehtmldom
    $html->load_file($sourceDomain); // the generated HTML
    $global_styles_css->load_file($sourceDomain . '/sites/all/themes/adaptivetheme/site_namedaptive/css/global.styles.css');
    $responsive_desktop_css->load_file($sourceDomain . '/sites/all/themes/adaptivetheme/site_namedaptive/css/responsive.desktop.css');


    // #header-wrapper and #nav-wrapper are the ids of the header and menu
    $topbar = $html->find('#header-wrapper', 0); // the zero here is necessary because simplehtmldom returns a collection
    $navbar = $html->find('#nav-wrapper', 0);

    // prepare Cross Origin Resource Sharing header
    $header = "<!-- This header necessary using CORS http://www.nczonline.net/blog/2010/05/25/cross-domain-ajax-with-cross-origin-resource-sharing/ -->
    <?php
    header('Access-Control-Allow-Origin: *');
    ?>";

    // some more CSS rules that are not in the css files
    $morestyles = '#google-appliance-block-form{width:100%;height:30px;text-align:right}#block-google-appliance-ga-block-search-form .container-inline div{display: inline;}.element-invisible, #block-google-appliance-ga-block-search-form label, .url-textfield{display:none!important;}#nav-wrapper ul li a, #menu-bar ul li a {padding: 5px 8px;}#nav-wrapper ul li.last, #menu-bar ul li.last {
    border-right: none;}';


    // prepare the CSS style tag; in case we do not want css, we only have to unset the $global_styles_css var
    $cssStyles = $global_styles_css ? '<style>' . $global_styles_css . $responsive_desktop_css . $morestyles . '</style>' : '';
    // replace relative image urls with absolute
    $cssStyles = str_ireplace("../",$sourceDomain . "/sites/all/themes/adaptivetheme/site_namedaptive/",$cssStyles);


    // assemble output from components
    $output = $header . $topbar . $navbar . $cssStyles;

        // create directory if not exists
        if (!file_exists('public://global_header')) {
                mkdir('public://global_header', 0777, true);
        }

    //write the file that will be fetched by the script tag embedded in the target site
        $filename = "public://global_header/raw_header.php";
    file_put_contents($filename,$output);
}

Here is my sample code - I am thinking about contributing a module for this function, with some admin settings

// get the header code and write it to a file
function module_name_global_header_write() {//dpm('raw_html');

    // there may be an issue if we are processing this function while viewing the same url as $sourceDomain
    // perhaps we will need to create a page or URL that is not viewed but can supply the HTML for the header
    $sourceDomain = 'http://site_name.com';
    $sourceMenuId = 'superfish-1';  

    // initialize simplehtmldom objects
    $html = new simple_html_dom();
    $responsive_desktop_css = new simple_html_dom();
    $global_styles_css = new simple_html_dom();

    // load files into simplehtmldom
    $html->load_file($sourceDomain); // the generated HTML
    $global_styles_css->load_file($sourceDomain . '/sites/all/themes/adaptivetheme/site_namedaptive/css/global.styles.css');
    $responsive_desktop_css->load_file($sourceDomain . '/sites/all/themes/adaptivetheme/site_namedaptive/css/responsive.desktop.css');


    // #header-wrapper and #nav-wrapper are the ids of the header and menu
    $topbar = $html->find('#header-wrapper', 0); // the zero here is necessary because simplehtmldom returns a collection
    $navbar = $html->find('#nav-wrapper', 0);

    // prepare Cross Origin Resource Sharing header
    $header = "<!-- This header necessary using CORS http://www.nczonline.net/blog/2010/05/25/cross-domain-ajax-with-cross-origin-resource-sharing/ -->
    <?php
    header('Access-Control-Allow-Origin: *');
    ?>";

    // some more CSS rules that are not in the css files
    $morestyles = '#google-appliance-block-form{width:100%;height:30px;text-align:right}#block-google-appliance-ga-block-search-form .container-inline div{display: inline;}.element-invisible, #block-google-appliance-ga-block-search-form label, .url-textfield{display:none!important;}#nav-wrapper ul li a, #menu-bar ul li a {padding: 5px 8px;}#nav-wrapper ul li.last, #menu-bar ul li.last {
    border-right: none;}';


    // prepare the CSS style tag; in case we do not want css, we only have to unset the $global_styles_css var
    $cssStyles = $global_styles_css ? '<style>' . $global_styles_css . $responsive_desktop_css . $morestyles . '</style>' : '';
    // replace relative image urls with absolute
    $cssStyles = str_ireplace("../",$sourceDomain . "/sites/all/themes/adaptivetheme/site_namedaptive/",$cssStyles);


    // assemble output from components
    $output = $header . $topbar . $navbar . $cssStyles;

        // create directory if not exists
        if (!file_exists('public://global_header')) {
                mkdir('public://global_header', 0777, true);
        }

    //write the file that will be fetched by the script tag embedded in the target site
    $filename = "public://global_header/raw_header.php";
    file_put_contents($filename,$output);
}

here is the code in the js file:

//cross-browser see http://www.nczonline.net/blog/2010/05/25/cross-domain-ajax-with-cross-origin-resource-sharing/
function createCORSRequest(method, url){
    var xhr = new XMLHttpRequest();
    if ("withCredentials" in xhr){
        xhr.open(method, url, true);
    } else if (typeof XDomainRequest != "undefined"){ //IE
        xhr = new XDomainRequest();
        xhr.open(method, url);
    } else {
        xhr = null;
    }
    return xhr;
}
var sourceURL = 'http://site.com/';
var request = createCORSRequest("get", sourceURL + "/sites/default/files/global_header/raw_header.php");

if (request){
    request.onload = function(){ 
    // do stuff if request is successful; if it fails, there is no replacement of existing HTML in target site
  // look first if a container with id #header-id exists, if not use the existing #topbar container
    document.getElementById('header-id') ? document.getElementById('header-id').innerHTML = request.responseText : document.getElementById('topbar').innerHTML = request.responseText;
    // change a relative image URL into absolute
    document.getElementById('branding').getElementsByTagName('img')[0].src = sourceURL + '/sites/default/files/image.png';
    // change relative Google Appliance Search action into absolute
    document.getElementById('google-appliance-block-form').action = sourceURL + '/';
    };
    request.send();
}

example script tag:

<script src="http://site.com/sites/all/modules/custom/module_name/module_name.js"></script>