There are a number of issues with the plan.
The biggest problem may be the routing of the water supply and drain across the door threshold. Wire covers are intended to cover wire, not hollow tubing. More importantly, the channels in them are usually not more than 3/4 high (most smaller) and will not fit a drain line. If you could find one large enough, it would pose a trip hazard at the doorway.
The next issue is the risk of damage to the tubing under the wire cover. You would need to ensure that the cover was rigid enough and stable enough to avoid both compression of the lines and tugging on the fittings (although you could put full or partial loops in the lines to avoid strain).
Dishwasher drain lines are usually fairly short, and the run you propose may be more than the dishwasher pump can handle. The hot water line is under pressure, so that would not be a problem, but I would consider using PEX to avoid crimping issues.
If you do this, you will still need to put in a branch on the drain and a branch with a shutoff valve on the water supply.
You may want to consider a portable dishwasher such as this type
Smaller ones are also available in countertop versions. They are wheeled to the sink for use, and attach to the faucet and drain over the lip of the sink. They can easily be taken with you and most can be converted to a built in when you are ready.