How can I change the value of a textbox whenever a dropdownlist within a gridview has its value changed?
On page load, the textbox shows the selected value, but when
I had a similar problem using the DropDownLists in GridView. My solution was to adjust the onLoad for the dropdown so that it wouldn't re-write the DropDownList on every post back. This way if there's something there then it won't re-populate it.
protected void dropDownLoad(object sender, EventArgs e)
{
    DropDownList dropDown = sender as DropDownList;
    if (dropDown.SelectedValue == null || dropDown.SelectedValue == "")
    { 
        // Your Code to populate table
    }
}
You need to use SelectedIndexChanged handler to show selected value:
Markup:
<asp:DropDownList ID="duty" runat="server" OnLoad="ddl1_load" OnSelectedIndexChanged="duty_SelectedIndexChanged"></asp:DropDownList>
Code-behind:
protected void duty_SelectedIndexChanged(object sender, EventArgs e)
{
    GridViewRow gvr = (GridViewRow)(((Control)sender).NamingContainer);   
    DropDownList duty= (DropDownList) gvr.FindControl("duty");
    TextBox1.Text = duty.SelectedItem.Value;
}
this happens to me once then i code like this... but i didnt use the onLoad attribute, tell me if this works,
 <asp:TemplateField HeaderText="duty" SortExpression="duty">
                                       <EditItemTemplate>
                                      <asp:TextBox ID="duty" runat="server" Text='<%# Bind("duty_Name") %>'></asp:TextBox>
                                    </EditItemTemplate>
                                      <ItemTemplate>
                                           <asp:Label ID="lblduty" runat="server" Text='<%# Eval("duty_Name") %>' />
                                        <asp:DropDownList ID="ddlduty" runat="server" CssClass="dropdownlist" 
                                          OnLoad = "ddl1_load" OnSelectedIndexChanged="duty_SelectedIndexChanged" Visible = "false" 
                                           >
                                        </asp:DropDownList>
                                    </ItemTemplate>
                                        <HeaderStyle Width="5%" />
                                        <ItemStyle HorizontalAlign="Center" />
                                    </asp:TemplateField>
You should look into using data binding instead. You can bind the textbox.Text to the selecteditem.value, this will ensure that proper updating takes place