procedure VCLTMemoHelper.GoToTextEnd;
begin
  PostMessage(Handle, WM_VSCROLL, SB_BOTTOM, 0);
end;

....
{$IFDEF HAS_FMX}
procedure MemoStringsDeleteLines(Memo : TMemo; LinesNum: Integer);
var
  MemoText,EndText : String;
  EndPos : Integer;
begin  //FMX cannot make replace for few Memo lines in 1 operation
  MemoText := Memo.Text;
  EndText := Memo.Lines[LinesNum];
  EndPos := PosEx_Sha_Pas_2(EndText,MemoText);
  MemoText := MemoText.Remove(0,EndPos + EndText.Length + Memo.Lines.LineBreak.Length - 1);
  Memo.Text := MemoText;
end;
{$ELSE}
procedure MemoStringsDeleteLines(Memo : TMemo; LinesNum: Integer);
var
  SelEnd: Integer;
begin
  SelEnd := SendMessage(Memo.Handle, EM_LINEINDEX, LinesNum, 0);
  if SelEnd < 0 then Exit;
  SendMessage(Memo.Handle, EM_SETSEL, 0, SelEnd);
  SendTextMessage(Memo.Handle, EM_REPLACESEL, 0, '');
end;
{$ENDIF}

...

procedure TBaseMainFormTransport.SysJournalUpdateTimerTimer(Sender : TObject);
begin
  if AppTerminated then
    Exit;

  SysJournalUpdateListSection.Enter;

  if SysJournalUpdateCount > 0 then
  begin
    StealthForm.SystemJournalMemo.Lines.BeginUpdate;

    StealthForm.SystemJournalMemo.Lines.Add(SysJournalUpdateStr);

    if StealthForm.SystemJournalMemo.Lines.Count > MaxSysJournalLines then
      MemoStringsDeleteLines(StealthForm.SystemJournalMemo,
                             StealthForm.SystemJournalMemo.Lines.Count - MaxSysJournalLines);

    SysJournalUpdateStr := '';
    SysJournalUpdateCount := 0;
    StealthForm.SystemJournalMemo.Lines.EndUpdate;
    {Перетягиваем ползунок сисжурнала в самый низ}
    StealthForm.SystemJournalMemo.GoToTextEnd;
  end;

  SysJournalUpdateListSection.Leave;
end;

...

procedure TBaseMainFormTransport.OnSystemJournalAdded(Sender : TObject; Value : String);
var time : String;      {Добавляем в SystemJournal из потока чара}
begin
  if AppTerminated then
    Exit;

  SysJournalUpdateListSection.Enter;

  time := FormatDateTime('hh:nn:ss:zzz', Now);

  if SysJournalUpdateCount = 0 then
    SysJournalUpdateStr := SysJournalUpdateStr + sLineBreak;
  SysJournalUpdateStr := SysJournalUpdateStr + time + ' [' + TBaseCharacter(Sender).ProfileName + ']: ' + Value;
  Inc(SysJournalUpdateCount);

  SysJournalUpdateListSection.Leave;
end;