This macro would go in the ThisWorkbook module and will watch for changes in column B of all sheets. If you change something, it will move it. 

Option Explicit

Private Sub Workbook_SheetChange(ByVal Sh As Object, ByVal Target As Range)
Dim cell As Range, wsNm As String

For Each cell In Target
    If cell.Column = 2 And cell.Value <> Sh.Name And cell.Value <> "" Then
        wsNm = cell.Value
        If Evaluate("ISREF('" & wsNm & "'!A1)") Then
            Application.EnableEvents = False
            cell.EntireRow.Cut Sheets(wsNm).Range("A" & Rows.Count).End(xlUp).Offset(1)
            Rows(cell.Row).Delete xlShiftUp
            Sheets(wsNm).Activate               'this is optional to follow the row...delete if not undesired
            Sheets(wsNm).Columns.AutoFit
            Application.EnableEvents = True
        Else
            MsgBox "No sheet with that name exists, row not moved."
        End If
    End If
Next cell

End Sub