When attempting to sort or filter data received via AJAX nothing happens except of getting a new request to the server. Even if try to trigger events such as: .trigger("update", [resort]) or .trigger("sorton", ...). If I use it without ajax it works perfectly.

$(function () {

    if (Modernizr.touch) {
        // make table cell focusable
        var $focus_highlight = $('.focus-highlight');
        if ($focus_highlight.length) {
            $focus_highlight
                .find('td, th')
                .attr('tabindex', '1')
                .on('touchstart', function () {
                    $(this).focus();
                });
        }
    }


    // list view
    altair_issues.list_view();
});

altair_issues = {
    list_view: function () {

        var $ts_issues = $("#ts_issues");

        // define pager options
        var pagerOptions = {
            // target the pager markup - see the HTML block below
            container: $(".ts_pager"),
            // output string - default is '{page}/{totalPages}'; possible variables: {page}, {totalPages}, {startRow}, {endRow} and {totalRows}
            output: '{startRow} - {endRow} / {filteredRows} ({totalRows})',
            // if true, the table will remain the same height no matter how many records are displayed. The space is made up by an empty
            // table row set to a height to compensate; default is false
            fixedHeight: true,
            // remove rows from the table to speed up the sort of large tables.
            // setting this to false, only hides the non-visible rows; needed if you plan to add/remove rows with the pager enabled.
            removeRows: false,
            // go to page selector - select dropdown that sets the current page
            cssGoto: '.ts_gotoPage',
            processAjaxOnInit: false,
            updateArrows: false,
            ajaxUrl: $ts_issues.data('ajax') + "?page={page}&size={size}",
            ajaxObject: {
                type: 'POST',
                data: getFormFilterData(),
                dataType: 'json',
                beforeSend: function (xhr, settings) {
                    taskListPreloader('open');
                    let QueryDataForSearch = $.param($('#tasks-filter-form').serializeArray());

                    // Второй вариант - '&TasksFilterForm%5BbShowFilters_%5D=' + 2 это игнорирования куки вообще!
                    let targetProject = ((new URL(window.location.href)).searchParams).get('target-project');
                    (targetProject === null) ? settings.data = QueryDataForSearch : settings.data = QueryDataForSearch += '&TasksFilterForm%5BbShowFilters_%5D=' + 2;
                }
            },
            ajaxProcessing: function (data, table, xhr) {
                taskListPreloader('close');
                if (data.status) {

                    let result = {
                        total: data.pager.total ?? 0
                    };

                    $.tablesorter.storage($('table'), 'tablesorter-pager', '');
                    console.log(data.html.tasks);
                    $(".tasks-list-table").html(data.html.tasks).trigger("update").tablesorterPager(pagerOptions);

                    if (data.description.length) {
                        bottomToast(data.description, "danger");
                    }

                    updatePagination(result.total);

                    return result;
                } else {
                    bottomToast(data.description, "danger");
                }
            },
            ajaxError: function (config, xhr, settings, exception) {
                console.log("Ajax Error:");
                console.log("Config:", config);
                console.log("XHR:", xhr);
                console.log("Settings:", settings);
                console.log("Exception:", exception);

                taskListPreloader('close');
            }
        };


        // Initialize tablesorter
        var ts_users = $ts_issues
            .tablesorter({
                theme: 'altair',
                widthFixed: true,
                widgets: ['zebra', 'filter'],
            })

As I sad I tried to trigger different events(not only from ajaxProcessing) but it didnt make sense. I also tried to disable serverSideSorting but it didnt work.