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.