Motor's of the electrical kind

Motors encompass a very large realm. I'm guessing you arent interested in dc motors, synchronous motors, 3 phase motors , etc,  just ordinary 1 phase induction motors? 

The reason a blower keeps spinning without power is its relatively large mechanical inertia compared to the frictional losses.

Pretty much all motors will keeping spinning without power as long as there is no torque being consumed by a load, or bearing friction. They will synch right back to a load dependent  full rpm when power is restored.

The motors with a capacitor start is for giving the motor a direction to spin when power is applied. The cap switches out after xxx rpm. 

Likely if you have a motor that goes to zero speed before restarting, a controller is making that decision, or it's got a hefty load or brake stopping it. 

I don't have experience with the various start systems for single phase motors, but dredging my memory I would think they all would continue running when power is re-applied (if they are still rotating), without needing the start switch. Permanent split capacitor motors, like fans always provide a rotating field so they don't need a start switch and for sure they will keep running. The switch start circuit is only used/needed to get motors initially rotating. Presumably the re-applied phase will align at some point as the motor slows down and get going again. A split phase motor even has a centrifugal switch on the rotor that disconnects the the start winding whenever the motor is rotating. Do you have a specific example of a motor that stops when power is re-applied? Is it under heavy load?

I will ask some questions to better understand your requirements.

You are looking for information on a motor configuration that has an inconsistent input power source or you are looking to control a motor by modulating the input power source such as pulse width modulation (PWM)?

Describe the motor configuration. Input parameters, load parameters and what the motor is driving?

Regarding: "disengage.. power engage the alternator and have the motor keep running "

Could you sketch it? Currently it sounds like a free energy thing.

An inconsistent power supply let's just say the grid drops in and drops out a lot but the motor needs to stay running even if it is only dropping in and out for a few seconds. The motor that I have currently When the power is disengaged something inside the motor clicks and it uses the electricity from back EMF to slow itself down and this is what I wished to bypass safely of course.

Motor theory is quite complex (at least, it is to me, it seemed to come easy for Tesla etc), but as far as I'm aware, it won't be back-EMF that is slowing the motor once any remaining magnetic field has collapsed after the grid has dropped out (which is a function of the windings), so I can't see what could be altered to make the motor continue to 'coast' longer than it currently is spinning for, which as you say isn't currently long enough for the switch to remain disengaged.
However, if the motor is powerful enough, perhaps a flywheel could be added, to store energy, allowing the motor to continue spinning for longer even when the power is temporarily removed. It likely can't be retrofitted if the motor is sized for no flywheel, and it could cause other problems, could even be dangerous.

What is the actual scenario/use-case? Perhaps there is a different solution (e.g. different motor would allow you to have a backup supply from a battery, etc).

The clicking noise you are likely hearing is the centrifugal switch on the cap winding dropping out.

The desired behavior is not a function of the motor, but the motor feeder. 

One way to do what you want is to have the shore power feed an inverter. The inverter drives the motor. The inverter would need a big capacitor or a battery bank on its dc link to provide the motor power while the ac feeder is switched. The feature is called power dip ride thru, but sounds like you want a beefier version for power loss ride thru. It's not a cheap solution, but can be done. 

Miracles can be engineered given infinite time and money :-) 

Tallboyshort  What you have described is called regenerative braking.  It's purpose in your systems is likely to minimize inductive generation and the uncontrolled voltage spikes that comes with it.  Inductive generation is the AC equivalent of the inductive spike created when de-energizing a DC coil.  If you were to defeat this system, I would HIGHLY encourage you to install a Transient Voltage Surge Suppression (TVSS) on your system.  The fact that is in the system makes me question if the safety shutdown - because that is what it is - can be safely bypassed.

Renererative braking is common in EVs and elevators.  Sometimes found in industrial machines.  Not common in commercial machines.  Almost unheard of in residential applications.  Again... it's an expensive adder so it's not there just for fun.

If your true problem is that your alternate power source cannot meet the inrush current demand of starting this piece of equipment, your better money is spend on a Soft Start (keep Torque Boost OFF) or a variable frequency drive.

Living above the arctic circle for a few years in the 80's, I have some experience with inconsistent power, Community AC Power was supplied by diesel generators. I had equipment at the very end of the transmission line. It had to live with any fallout from downstream switching. Five in the afternoon when cooking stoves got turned on, left me with almost a 10V drop in AC.

What do you have for a solution budget? When you have exhausted all the avenues through the power supplier it usually means dipping into your own pocket to pay for a solution that conditions the AC power for your use.

My initial thought is an inline UPS or some type of automatic voltage regulation. I have worked in areas that have both. Unfortunately the solutions are not cheap. The environment was for IT systems so clean power was critical.