How do we connect in Cyprus to our flight with Emirates?

I am travelling with my three children from Beirut to Malta, with a 3-hour layover in Larnaca before a connecting Emirates flight. The flights are booked separately

If the flights are booked separately then they are not connecting flights. They are simply two separate flights. It is up to you to take yourselves and your bags (and your children) between those two, unconnected, flights.

If you have checked-in baggage which is not through-checked, you will need to go landside to collect your bags, proceed through customs and passport control and then take your bags through to the Emirates check in desk. Cyprus is not a member of the Schengen agreement so you may need a separate visa for this compared to entering Malta. [Although I don't think Larnaca can actually handle airside connections anyway?--Comments welcome.]

In the event you miss your onward flight you will be considered a "no show" and (unless your Emirates ticket is flexible) you may lose the value of the Emirates ticket, including any onward or return sectors later on the same ticket.

The flights are booked separately and there is no interline agreements between our first carrier, Middle East Airlines (MEA), and Emirates.

According to my information, Middle East Airlines (ME) does have a forwarding agreement with Emirates (EK):

 MAY CHECK BAGGAGE TO
   AA  AC  AE  AF  AH  AI  AM  AR  AT  AV  AZ  A3  BA  BE  
   CA  CI  CU  CX  CZ  DL  D6  EK  ET  EW  EY  FI  FM  GA  
   GF  G3  HR  IC  IG  IR  JJ  JL  JU  KC  KE  KK  KL  KM  
   KU  KW  LH  LO  LR  LX  LZ  MD  ME  MF  MS  MU  NF  OA  
   OK  OU  OV  OZ  PK  PX  PZ  QE  QR  RB  RO  SK  SN  SQ  
   SU  SV  TA  TG  TK  TN  TP  TU  UA  UL  UU  U8  VA  VN  
   VS  WA  WF  WY  XM  YO  2A  9W

Although it is unlikely, it may be possible to through-check your baggage to the Emirates flight at your initial check in, by presenting all the relevant documents for your Emirates flight. However, you cannot rely on this: it depends on the training of the check in agent and their willingness to bend the rules.