Wolfster,
A player wants in, you deal him in. Cheques (chips) coming from the dealer is simple, but when a chip runner is needed, it is a priority to get back to the table with the players cheques ASAP. If the players chips are not back in time, he usually borrows a stack from a neighbor until the runner returns, (It's not something that we advertise, but every poker room does it), or the dealer will take the amount that is owed from the pot and keep it in front of the waiting player. The player is "light" that amount. When his cheques arrive he will deposit what he owes and play continues.
I'll give you an example of what might occur at a table. The game is limit hold'em $3 and $6. The min buy-in is $30. Player on the BB goes all-in and loses. He reaches into his pocket and takes out a $100 bill, tosses it to the dealer and says, deal me in," I'd like $40 in chips, please." If the dealer can make his change, he will. If the dealer is not able to change the $100 he will call out to the chip runner, or whoever is responsible, "PLAYER'S CHEQUES ON TABLE 16, FORTY WHITE AND SIXTY SOFT!" When the runner returns the player must put $60 back in his pocket. The methods I've mentioned are designed to keep the game going without slowing it down too much.
While I was typing K-Lo offered his excellent examples. I don't know if this is practiced anymore but we used to play "on the piece" back in the old days, that is, we would take a large bill like a $100 and drag money from the pot and put it on the bill to indicate how much was owed. That was a bad practice because there were times that players would fold and take back their $100 bill along with the $30 in cash still on the piece!
Not a recommended practice. I will say that it was always done in house games back then.