In　this　paper,　we　first　consider　a　dynamic　k-level　facility　location　problem,　which　is　a　generalization　of　the　k-level　facility　location　problem　when　considering　time　factor.　We　present　a　combinatorial　primal-dual　approximation　algorithm　for　this　problem　which　finds　a　constant　factor　approximate　solution.　Then,　we　investigative　the　dynamic　k-level　facility　location　problem　with　submodular　penalties　and　outliers,　which　extend　the　existing　problem　on　two　fronts,　namely　from　static　to　dynamic　and　from　without　penalties　(outliers)　to　penalties　(outliers)　allowed.　Based　on　primal-dual　technique　and　the　triangle　inequality　property,　we　also　give　two　constant　factor　approximation　algorithms　for　the　dynamic　problem　with　submodular　penalties　and　outliers,　respectively.　©　2020　Elsevier　B.V.