I'm no plumber, but I've had my share of washing machine lint clogs. A washing machine puts out a LOT of water at one time, and that water comes with plenty of lint.
A typical setup (but not the only way to do it) is to have the output from the washing machine sit in a laundry tub. That (a) provides a buffer of sorts - if the main drain can't take the water that fast the tub can hold some of it as it drains out and (b) because the end of the hose is accessible you can attach a disposable metal mesh lint trap. I don't recommend trying the same thing with the kitchen sink - you will likely end up with the sink overflowing if the drain isn't 100% clear at the time, as most sinks are far smaller than a laundry tub. On the other hand, if you connect the hose directly into the drain line (with an appropriate trap AFTER the connection), you will be putting lint directly into the same section of pipe that is routinely getting grease and food from the sink.
I actually don't think the size of the drain hose itself will be a problem - my dishwasher drain hoses are similar size to my washing machine drain hose (though different materials) - the issues are the total volume of water in a very short amount of time and potential lint accumulation.