function terminar_tramite(ids) {

    var ID = ids;
    console.log(ID);
    $.ajax({
        url: "lib/controladores/tramite_terminado.php",
        type: 'POST',
        data: `idss=${ID}`,
        contentType: false,
        processData: false,
        success: function(response){
            if (response.success) {
               console.log("no está guardando")
                UIkit.notification({
                    message: 'Ocurrió un error!',
                    status: 'primary',
                    pos: 'top-right',
                    timeout: 5000
                });
            }else{
                UIkit.notification({
                    message: 'Tramite Guardado!',
                    status: 'success',
                    pos: 'bottom-center',
                    timeout: 5000
                });
                setTimeout(function(){
                    location.href = `?sub=adm&op=tablero_tramites`;
                },2000);
                setTimeout(function(){
                location.href = `?sub=adm&op=tablero_tramites`;
                },2000);
            }
        }
    })
}

<?php
    include '../../config.php';
    $dbserver = DB_SERVER;
    $dbuser = DB_USER;
    $password = DB_PASS;
    $dbname = DB_NAME;
    $database = new mysqli($dbserver, $dbuser, $password, $dbname);
    $variable = $_POST['idss'];
    var_dump($variable);
    if($database->connect_errno) {
        die("No se pudo conectar a la base de datos");
    }
    $insertquery  = " INSERT INTO tramites_atendidos SELECT * FROM tramites WHERE tr_id = '$variable'";
    $jsondata = array();
    if($database->query($insertquery)){
    }
    else {
    }
    $database->close();
?>

data: {
   "idss": ID
},

function terminar_tramite(ids) {

    var ID = ids;
    console.log(ID);
    $.ajax({
        url: "lib/controladores/tramite_terminado.php",
        type: 'POST',
        data: ID, //Solo cambiar esta linea
        contentType: false,
        processData: false,
        success: function(response){
            if (response.success) {
               console.log("no está guardando")
                UIkit.notification({
                    message: 'Ocurrió un error!',
                    status: 'primary',
                    pos: 'top-right',
                    timeout: 5000
                });
            }else{
                UIkit.notification({
                    message: 'Tramite Guardado!',
                    status: 'success',
                    pos: 'bottom-center',
                    timeout: 5000
                });
                setTimeout(function(){
                    location.href = `?sub=adm&op=tablero_tramites`;
                },2000);
                setTimeout(function(){
                location.href = `?sub=adm&op=tablero_tramites`;
                },2000);
            }
        }
    })
}