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Isometric character movement, traversing nodes

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I have a question regarding character movement in an isometric turn based strategy game, characters move from tile to tile only. As of now, I'm using linear interpolation to move the character from tile to tile, but that causes a problem: the character lags when switching movement.

Did some research, and found out, that I should be using "Piecewise linear interpolation" - how does it work in the real world? I found articles about it, but they weren't so easy to understand, mostly because they are very mathematical and English is not my native tongue. Could someone please describe it more understandably?

For now, I'm using the following code to traverse between two nodes:

void Character::MovementUpdate(){
        float rate=(float)(moveTime-movesLeft)/moveTime;
        posX=(int) ( (float)(currentTile->x+20)*rate + (float)posX*((float)1-rate));
        posY=(int) ( (float)(currentTile->y-25)*rate + (float)posY*((float)1-rate));


If the character is moving, calculate it's coordinates. There are 30 steps for the character to move from A to B, and if there are more nodes to go to, go to the next node. For path-finding A* is being used, which returns a list of moves the character has to make, for example: North, North, West; etc.

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